Английская Википедия:Continuous function
Шаблон:Short description Шаблон:Calculus
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is Шаблон:Em. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.
Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology.
A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity.
As an example, the function Шаблон:Math denoting the height of a growing flower at time Шаблон:Mvar would be considered continuous. In contrast, the function Шаблон:Math denoting the amount of money in a bank account at time Шаблон:Mvar would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.
History
A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of <math>y = f(x)</math> as follows: an infinitely small increment <math>\alpha</math> of the independent variable x always produces an infinitely small change <math>f(x+\alpha)-f(x)</math> of the dependent variable y (see e.g. Cours d'Analyse, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano,[1] Karl Weierstrass[2] denied continuity of a function at a point c unless it was defined at and on both sides of c, but Édouard Goursat[3] allowed the function to be defined only at and on one side of c, and Camille Jordan[4] allowed it even if the function was defined only at c. All three of those nonequivalent definitions of pointwise continuity are still in use.[5] Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.[6]
Real functions
Definition
A real function that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below.[8]
Continuity of real functions is usually defined in terms of limits. A function Шаблон:Math with variable Шаблон:Mvar is continuous at the real number Шаблон:Mvar, if the limit of <math>f(x),</math> as Шаблон:Mvar tends to Шаблон:Mvar, is equal to <math>f(c).</math>
There are several different definitions of the (global) continuity of a function, which depend on the nature of its domain.
A function is continuous on an open interval if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval <math>(-\infty, +\infty)</math> (the whole real line) is often called simply a continuous function; one also says that such a function is continuous everywhere. For example, all polynomial functions are continuous everywhere.
A function is continuous on a semi-open or a closed interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function <math>f(x) = \sqrt{x}</math> is continuous on its whole domain, which is the closed interval <math>[0,+\infty).</math>
Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points. Examples are the functions <math display="inline">x \mapsto \frac {1}{x}</math> and <math>x\mapsto \tan x.</math> When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous.
A partial function is discontinuous at a point if the point belongs to the topological closure of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions <math display="inline">x\mapsto \frac {1}{x}</math> and <math display="inline">x\mapsto \sin(\frac {1}{x})</math> are discontinuous at Шаблон:Math, and remain discontinuous whichever value is chosen for defining them at Шаблон:Math. A point where a function is discontinuous is called a discontinuity.
Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.
Let <math display="block">f : D \to \R</math> be a function defined on a subset <math>D</math> of the set <math>\R</math> of real numbers.
This subset <math>D</math> is the domain of Шаблон:Math. Some possible choices include
- <math>D = \R </math>: i.e., <math> D </math> is the whole set of real numbers. or, for Шаблон:Mvar and Шаблон:Mvar real numbers,
- <math>D = [a, b] = \{x \in \R \mid a \leq x \leq b \} </math>: <math> D </math> is a closed interval, or
- <math>D = (a, b) = \{x \in \R \mid a < x < b \} </math>: <math> D </math> is an open interval.
In the case of the domain <math>D</math> being defined as an open interval, <math>a</math> and <math>b</math> do not belong to <math>D</math>, and the values of <math>f(a)</math> and <math>f(b)</math> do not matter for continuity on <math>D</math>.
Definition in terms of limits of functions
The function Шаблон:Math is continuous at some point Шаблон:Math of its domain if the limit of <math>f(x),</math> as x approaches c through the domain of f, exists and is equal to <math>f(c).</math>[9] In mathematical notation, this is written as <math display="block">\lim_{x \to c}{f(x)} = f(c).</math> In detail this means three conditions: first, Шаблон:Math has to be defined at Шаблон:Math (guaranteed by the requirement that Шаблон:Math is in the domain of Шаблон:Math). Second, the limit of that equation has to exist. Third, the value of this limit must equal <math>f(c).</math>
(Here, we have assumed that the domain of f does not have any isolated points.)
Definition in terms of neighborhoods
A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point <math>f(c)</math> as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood <math>N_1(f(c))</math> there is a neighborhood <math>N_2(c)</math> in its domain such that <math>f(x) \in N_1(f(c))</math> whenever <math>x\in N_2(c).</math>
As neighborhoods are defined in any topological space, this definition of a continuous function applies not only for real functions but also when the domain and the codomain are topological spaces and is thus the most general definition. It follows that a function is automatically continuous at every isolated point of its domain. For example, every real-valued function on the integers is continuous.
Definition in terms of limits of sequences
One can instead require that for any sequence <math>(x_n)_{n \in \N}</math> of points in the domain which converges to c, the corresponding sequence <math>\left(f(x_n)\right)_{n\in \N}</math> converges to <math>f(c).</math> In mathematical notation, <math display="block">\forall (x_n)_{n \in \N} \subset D:\lim_{n\to\infty} x_n = c \Rightarrow \lim_{n\to\infty} f(x_n) = f(c)\,.</math>
Weierstrass and Jordan definitions (epsilon–delta) of continuous functions
Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function <math>f : D \to \mathbb{R}</math> as above and an element <math>x_0</math> of the domain <math>D</math>, <math>f</math> is said to be continuous at the point <math>x_0</math> when the following holds: For any positive real number <math>\varepsilon > 0,</math> however small, there exists some positive real number <math>\delta > 0</math> such that for all <math>x</math> in the domain of <math>f</math> with <math>x_0 - \delta < x < x_0 + \delta,</math> the value of <math>f(x)</math> satisfies <math display="block">f\left(x_0\right) - \varepsilon < f(x) < f(x_0) + \varepsilon.</math>
Alternatively written, continuity of <math>f : D \to \mathbb{R}</math> at <math>x_0 \in D</math> means that for every <math>\varepsilon > 0,</math> there exists a <math>\delta > 0</math> such that for all <math>x \in D</math>: <math display="block">\left|x - x_0\right| < \delta ~~\text{ implies }~~ |f(x) - f(x_0)| < \varepsilon.</math>
More intuitively, we can say that if we want to get all the <math>f(x)</math> values to stay in some small neighborhood around <math>f\left(x_0\right),</math> we need to choose a small enough neighborhood for the <math>x</math> values around <math>x_0.</math> If we can do that no matter how small the <math>f(x_0)</math> neighborhood is, then <math>f</math> is continuous at <math>x_0.</math>
In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the metric topology.
Weierstrass had required that the interval <math>x_0 - \delta < x < x_0 + \delta</math> be entirely within the domain <math>D</math>, but Jordan removed that restriction.
Definition in terms of control of the remainder
In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. A function <math>C: [0,\infty) \to [0,\infty]</math> is called a control function if
- C is non-decreasing
- <math>\inf_{\delta > 0} C(\delta) = 0</math>
A function <math>f : D \to R</math> is C-continuous at <math>x_0</math> if there exists such a neighbourhood <math display="inline">N(x_0)</math> that <math display="block">|f(x) - f(x_0)| \leq C\left(\left|x - x_0\right|\right) \text{ for all } x \in D \cap N(x_0)</math>
A function is continuous in <math>x_0</math> if it is C-continuous for some control function C.
This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions <math>\mathcal{C}</math> a function is Шаблон:Nowrap if it is Шаблон:Nowrap for some <math>C \in \mathcal{C}.</math> For example, the Lipschitz and Hölder continuous functions of exponent Шаблон:Mvar below are defined by the set of control functions <math display="block">\mathcal{C}_{\mathrm{Lipschitz}} = \{C : C(\delta) = K|\delta| ,\ K > 0\}</math> respectively <math display="block">\mathcal{C}_{\text{Hölder}-\alpha} = \{C : C(\delta) = K |\delta|^\alpha, \ K > 0\}.</math>
Definition using oscillation
Continuity can also be defined in terms of oscillation: a function f is continuous at a point <math>x_0</math> if and only if its oscillation at that point is zero;[10] in symbols, <math>\omega_f(x_0) = 0.</math> A benefit of this definition is that it Шаблон:Em discontinuity: the oscillation gives how Шаблон:Em the function is discontinuous at a point.
This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than <math>\varepsilon</math> (hence a <math>G_{\delta}</math> set) – and gives a rapid proof of one direction of the Lebesgue integrability condition.[11]
The oscillation is equivalent to the <math>\varepsilon-\delta</math> definition by a simple re-arrangement and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given <math>\varepsilon_0</math> there is no <math>\delta</math> that satisfies the <math>\varepsilon-\delta</math> definition, then the oscillation is at least <math>\varepsilon_0,</math> and conversely if for every <math>\varepsilon</math> there is a desired <math>\delta,</math> the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.
Definition using the hyperreals
Cauchy defined the continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34). Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.
(see microcontinuity). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's definition of continuity.
Construction of continuous functions
Checking the continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Given <math display="block">f, g \colon D \to \R,</math> then the Шаблон:Em <math display="block">s = f + g</math> (defined by <math>s(x) = f(x) + g(x)</math> for all <math>x\in D</math>) is continuous in <math>D.</math>
The same holds for the Шаблон:Em, <math display="block">p = f \cdot g</math> (defined by <math>p(x) = f(x) \cdot g(x)</math> for all <math>x \in D</math>) is continuous in <math>D.</math>
Combining the above preservations of continuity and the continuity of constant functions and of the identity function <math>I(x) = x</math> Шаблон:Nowrap one arrives at the continuity of all polynomial functions Шаблон:Nowrap such as <math display="block">f(x) = x^3 + x^2 - 5 x + 3</math> (pictured on the right).
In the same way, it can be shown that the Шаблон:Em <math display="block">r = 1/f</math> (defined by <math>r(x) = 1/f(x)</math> for all <math>x \in D</math> such that <math>f(x) \neq 0</math>) is continuous in <math>D\setminus \{x : f(x) = 0\}.</math>
This implies that, excluding the roots of <math>g,</math> the Шаблон:Em <math display="block">q = f / g</math> (defined by <math>q(x) = f(x)/g(x)</math> for all <math>x \in D</math>, such that <math>g(x) \neq 0</math>) is also continuous on <math>D\setminus \{x:g(x) = 0\}</math>.
For example, the function (pictured) <math display="block">y(x) = \frac{2x-1}{x+2}</math> is defined for all real numbers <math>x \neq -2</math> and is continuous at every such point. Thus, it is a continuous function. The question of continuity at <math>x = -2</math> does not arise since <math>x = -2</math> is not in the domain of <math>y.</math> There is no continuous function <math>F : \R \to \R</math> that agrees with <math>y(x)</math> for all <math>x \neq -2.</math>
Since the function sine is continuous on all reals, the sinc function <math>G(x) = \sin(x)/x,</math> is defined and continuous for all real <math>x \neq 0.</math> However, unlike the previous example, G Шаблон:Em be extended to a continuous function on Шаблон:Em real numbers, by Шаблон:Em the value <math>G(0)</math> to be 1, which is the limit of <math>G(x),</math> when x approaches 0, i.e., <math display="block">G(0) = \lim_{x\to 0} \frac{\sin x}{x} = 1.</math>
Thus, by setting
- <math>
G(x) = \begin{cases} \frac {\sin (x)}x & \text{ if }x \ne 0\\ 1 & \text{ if }x = 0, \end{cases} </math>
the sinc-function becomes a continuous function on all real numbers. The term Шаблон:Em is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points.
A more involved construction of continuous functions is the function composition. Given two continuous functions <math display="block">g : D_g \subseteq \R \to R_g \subseteq \R \quad \text{ and } \quad f : D_f \subseteq \R \to R_f \subseteq D_g,</math> their composition, denoted as <math>c = g \circ f : D_f \to \R,</math> and defined by <math>c(x) = g(f(x)),</math> is continuous.
This construction allows stating, for example, that <math display="block">e^{\sin(\ln x)}</math> is continuous for all <math>x > 0.</math>
Examples of discontinuous functions
An example of a discontinuous function is the Heaviside step function <math>H</math>, defined by <math display="block">H(x) = \begin{cases} 1 & \text{ if } x \ge 0\\ 0 & \text{ if } x < 0 \end{cases} </math>
Pick for instance <math>\varepsilon = 1/2</math>. Then there is no Шаблон:Nowrap around <math>x = 0</math>, i.e. no open interval <math>(-\delta,\;\delta)</math> with <math>\delta > 0,</math> that will force all the <math>H(x)</math> values to be within the Шаблон:Nowrap of <math>H(0)</math>, i.e. within <math>(1/2,\;3/2)</math>. Intuitively, we can think of this type of discontinuity as a sudden jump in function values.
Similarly, the signum or sign function <math display="block"> \sgn(x) = \begin{cases} \;\;\ 1 & \text{ if }x > 0\\ \;\;\ 0 & \text{ if }x = 0\\ -1 & \text{ if }x < 0 \end{cases} </math> is discontinuous at <math>x = 0</math> but continuous everywhere else. Yet another example: the function <math display="block">f(x) = \begin{cases}
\sin\left(x^{-2}\right)&\text{ if }x \neq 0\\
0&\text{ if }x = 0
\end{cases}</math> is continuous everywhere apart from <math>x = 0</math>.
Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined pathological, for example, Thomae's function, <math display="block">f(x)=\begin{cases} 1 &\text{ if } x=0\\ \frac{1}{q}&\text{ if } x = \frac{p}{q} \text{(in lowest terms) is a rational number}\\
0&\text{ if }x\text{ is irrational}.
\end{cases}</math> is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, Dirichlet's function, the indicator function for the set of rational numbers, <math display="block">D(x)=\begin{cases}
0&\text{ if }x\text{ is irrational } (\in \R \setminus \Q)\\
1&\text{ if }x\text{ is rational } (\in \Q)
\end{cases}</math> is nowhere continuous.
Properties
A useful lemma
Let <math>f(x)</math> be a function that is continuous at a point <math>x_0,</math> and <math>y_0</math> be a value such <math>f\left(x_0\right)\neq y_0.</math> Then <math>f(x)\neq y_0</math> throughout some neighbourhood of <math>x_0.</math>[12]
Proof: By the definition of continuity, take <math>\varepsilon =\frac{|y_0-f(x_0)|}{2}>0</math> , then there exists <math>\delta>0</math> such that <math display="block">\left|f(x)-f(x_0)\right| < \frac{\left|y_0 - f(x_0)\right|}{2} \quad \text{ whenever } \quad |x-x_0| < \delta</math> Suppose there is a point in the neighbourhood <math>|x-x_0|<\delta</math> for which <math>f(x)=y_0;</math> then we have the contradiction <math display="block">\left|f(x_0)-y_0\right| < \frac{\left|f(x_0) - y_0\right|}{2}.</math>
Intermediate value theorem
The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states:
- If the real-valued function f is continuous on the closed interval <math>[a, b],</math> and k is some number between <math>f(a)</math> and <math>f(b),</math> then there is some number <math>c \in [a, b],</math> such that <math>f(c) = k.</math>
For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.
As a consequence, if f is continuous on <math>[a, b]</math> and <math>f(a)</math> and <math>f(b)</math> differ in sign, then, at some point <math>c \in [a, b],</math> <math>f(c)</math> must equal zero.
Extreme value theorem
The extreme value theorem states that if a function f is defined on a closed interval <math>[a, b]</math> (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists <math>c \in [a, b]</math> with <math>f(c) \geq f(x)</math> for all <math>x \in [a, b].</math> The same is true of the minimum of f. These statements are not, in general, true if the function is defined on an open interval <math>(a, b)</math> (or any set that is not both closed and bounded), as, for example, the continuous function <math>f(x) = \frac{1}{x},</math> defined on the open interval (0,1), does not attain a maximum, being unbounded above.
Relation to differentiability and integrability
Every differentiable function <math display="block">f : (a, b) \to \R</math> is continuous, as can be shown. The converse does not hold: for example, the absolute value function
- <math>f(x)=|x| = \begin{cases}
\;\;\ x & \text{ if }x \geq 0\\
-x & \text{ if }x < 0
\end{cases}</math> is everywhere continuous. However, it is not differentiable at <math>x = 0</math> (but is so everywhere else). Weierstrass's function is also everywhere continuous but nowhere differentiable.
The derivative f′(x) of a differentiable function f(x) need not be continuous. If f′(x) is continuous, f(x) is said to be continuously differentiable. The set of such functions is denoted <math>C^1((a, b)).</math> More generally, the set of functions <math display="block">f : \Omega \to \R</math> (from an open interval (or open subset of <math>\R</math>) <math>\Omega</math> to the reals) such that f is <math>n</math> times differentiable and such that the <math>n</math>-th derivative of f is continuous is denoted <math>C^n(\Omega).</math> See differentiability class. In the field of computer graphics, properties related (but not identical) to <math>C^0, C^1, C^2</math> are sometimes called <math>G^0</math> (continuity of position), <math>G^1</math> (continuity of tangency), and <math>G^2</math> (continuity of curvature); see Smoothness of curves and surfaces.
Every continuous function <math display="block">f : [a, b] \to \R</math> is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable but discontinuous) sign function shows.
Pointwise and uniform limits
Given a sequence <math display="block">f_1, f_2, \dotsc : I \to \R</math> of functions such that the limit <math display="block">f(x) := \lim_{n \to \infty} f_n(x)</math> exists for all <math>x \in D,</math>, the resulting function <math>f(x)</math> is referred to as the pointwise limit of the sequence of functions <math>\left(f_n\right)_{n \in N}.</math> The pointwise limit function need not be continuous, even if all functions <math>f_n</math> are continuous, as the animation at the right shows. However, f is continuous if all functions <math>f_n</math> are continuous and the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the exponential functions, logarithms, square root function, and trigonometric functions are continuous.
Directional and semi-continuity
-
A right-continuous function
-
A left-continuous function
Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is Шаблон:Em if no jump occurs when the limit point is approached from the right. Formally, f is said to be right-continuous at the point c if the following holds: For any number <math>\varepsilon > 0</math> however small, there exists some number <math>\delta > 0</math> such that for all x in the domain with <math>c < x < c + \delta,</math> the value of <math>f(x)</math> will satisfy <math display="block">|f(x) - f(c)| < \varepsilon.</math>
This is the same condition as continuous functions, except it is required to hold for x strictly larger than c only. Requiring it instead for all x with <math>c - \delta < x < c</math> yields the notion of Шаблон:Em functions. A function is continuous if and only if it is both right-continuous and left-continuous.
A function f is Шаблон:Em if, roughly, any jumps that might occur only go down, but not up. That is, for any <math>\varepsilon > 0,</math> there exists some number <math>\delta > 0</math> such that for all x in the domain with <math>|x - c| < \delta,</math> the value of <math>f(x)</math> satisfies <math display="block">f(x) \geq f(c) - \epsilon.</math> The reverse condition is Шаблон:Em.
Continuous functions between metric spaces
The concept of continuous real-valued functions can be generalized to functions between metric spaces. A metric space is a set <math>X</math> equipped with a function (called metric) <math>d_X,</math> that can be thought of as a measurement of the distance of any two elements in X. Formally, the metric is a function <math display="block">d_X : X \times X \to \R</math> that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces <math>\left(X, d_X\right)</math> and <math>\left(Y, d_Y\right)</math> and a function <math display="block">f : X \to Y</math> then <math>f</math> is continuous at the point <math>c \in X</math> (with respect to the given metrics) if for any positive real number <math>\varepsilon > 0,</math> there exists a positive real number <math>\delta > 0</math> such that all <math>x \in X</math> satisfying <math>d_X(x, c) < \delta</math> will also satisfy <math>d_Y(f(x), f(c)) < \varepsilon.</math> As in the case of real functions above, this is equivalent to the condition that for every sequence <math>\left(x_n\right)</math> in <math>X</math> with limit <math>\lim x_n = c,</math> we have <math>\lim f\left(x_n\right) = f(c).</math> The latter condition can be weakened as follows: <math>f</math> is continuous at the point <math>c</math> if and only if for every convergent sequence <math>\left(x_n\right)</math> in <math>X</math> with limit <math>c</math>, the sequence <math>\left(f\left(x_n\right)\right)</math> is a Cauchy sequence, and <math>c</math> is in the domain of <math>f</math>.
The set of points at which a function between metric spaces is continuous is a <math>G_{\delta}</math> set – this follows from the <math>\varepsilon-\delta</math> definition of continuity.
This notion of continuity is applied, for example, in functional analysis. A key statement in this area says that a linear operator <math display="block">T : V \to W</math> between normed vector spaces <math>V</math> and <math>W</math> (which are vector spaces equipped with a compatible norm, denoted <math>\|x\|</math>) is continuous if and only if it is bounded, that is, there is a constant <math>K</math> such that <math display="block">\|T(x)\| \leq K \|x\|</math> for all <math>x \in V.</math>
Uniform, Hölder and Lipschitz continuity
The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way <math>\delta</math> depends on <math>\varepsilon</math> and c in the definition above. Intuitively, a function f as above is uniformly continuous if the <math>\delta</math> does not depend on the point c. More precisely, it is required that for every real number <math>\varepsilon > 0</math> there exists <math>\delta > 0</math> such that for every <math>c, b \in X</math> with <math>d_X(b, c) < \delta,</math> we have that <math>d_Y(f(b), f(c)) < \varepsilon.</math> Thus, any uniformly continuous function is continuous. The converse does not generally hold but holds when the domain space X is compact. Uniformly continuous maps can be defined in the more general situation of uniform spaces.[13]
A function is Hölder continuous with exponent α (a real number) if there is a constant K such that for all <math>b, c \in X,</math> the inequality <math display="block">d_Y (f(b), f(c)) \leq K \cdot (d_X (b, c))^\alpha</math> holds. Any Hölder continuous function is uniformly continuous. The particular case <math>\alpha = 1</math> is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant K such that the inequality <math display="block">d_Y (f(b), f(c)) \leq K \cdot d_X (b, c)</math> holds for any <math>b, c \in X.</math>[14] The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.
Шаблон:AnchorContinuous functions between topological spaces
Another, more abstract, notion of continuity is the continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set X together with a topology on X, which is a set of subsets of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing one to talk about the neighborhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology).
A function <math display="block">f : X \to Y</math> between two topological spaces X and Y is continuous if for every open set <math>V \subseteq Y,</math> the inverse image <math display="block">f^{-1}(V) = \{x \in X \; | \; f(x) \in V \}</math> is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology <math>T_X</math>), but the continuity of f depends on the topologies used on X and Y.
This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X.
An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions <math display="block">f : X \to T</math> to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X) and the space T set is at least T0, then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.
Continuity at a point
The translation in the language of neighborhoods of the <math>(\varepsilon, \delta)</math>-definition of continuity leads to the following definition of the continuity at a point: Шаблон:Quote frame
This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using preimages rather than images.
Also, as every set that contains a neighborhood is also a neighborhood, and <math>f^{-1}(V)</math> is the largest subset Шаблон:Mvar of Шаблон:Mvar such that <math>f(U) \subseteq V,</math> this definition may be simplified into: Шаблон:Quote frame
As an open set is a set that is a neighborhood of all its points, a function <math>f : X \to Y</math> is continuous at every point of Шаблон:Mvar if and only if it is a continuous function.
If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f(x) instead of all neighborhoods. This gives back the above <math>\varepsilon-\delta</math> definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If, however, the target space is a Hausdorff space, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.
Given <math>x \in X,</math> a map <math>f : X \to Y</math> is continuous at <math>x</math> if and only if whenever <math>\mathcal{B}</math> is a filter on <math>X</math> that converges to <math>x</math> in <math>X,</math> which is expressed by writing <math>\mathcal{B} \to x,</math> then necessarily <math>f(\mathcal{B}) \to f(x)</math> in <math>Y.</math> If <math>\mathcal{N}(x)</math> denotes the neighborhood filter at <math>x</math> then <math>f : X \to Y</math> is continuous at <math>x</math> if and only if <math>f(\mathcal{N}(x)) \to f(x)</math> in <math>Y.</math>Шаблон:Sfn Moreover, this happens if and only if the prefilter <math>f(\mathcal{N}(x))</math> is a filter base for the neighborhood filter of <math>f(x)</math> in <math>Y.</math>Шаблон:Sfn
Alternative definitions
Several equivalent definitions for a topological structure exist; thus, several equivalent ways exist to define a continuous function.
Sequences and nets Шаблон:Anchor
In several contexts, the topology of a space is conveniently specified in terms of limit points. This is often accomplished by specifying when a point is the limit of a sequence. Still, for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.
In detail, a function <math>f : X \to Y</math> is sequentially continuous if whenever a sequence <math>\left(x_n\right)</math> in <math>X</math> converges to a limit <math>x,</math> the sequence <math>\left(f\left(x_n\right)\right)</math> converges to <math>f(x).</math> Thus, sequentially continuous functions "preserve sequential limits." Every continuous function is sequentially continuous. If <math>X</math> is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if <math>X</math> is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve the limits of nets, and this property characterizes continuous functions.
For instance, consider the case of real-valued functions of one real variable:[15]
Шаблон:Math theorem Шаблон:Collapse top Proof. Assume that <math>f : A \subseteq \R \to \R</math> is continuous at <math>x_0</math> (in the sense of <math>\epsilon-\delta</math> continuity). Let <math>\left(x_n\right)_{n\geq1}</math> be a sequence converging at <math>x_0</math> (such a sequence always exists, for example, <math>x_n = x, \text{ for all } n</math>); since <math>f</math> is continuous at <math>x_0</math> <math display=block>\forall \epsilon > 0\, \exists \delta_{\epsilon} > 0 : 0 < |x-x_0| < \delta_{\epsilon} \implies |f(x)-f(x_0)| < \epsilon.\quad (*)</math> For any such <math>\delta_{\epsilon}</math> we can find a natural number <math>\nu_{\epsilon} > 0</math> such that for all <math>n > \nu_{\epsilon},</math> <math display=block>|x_n-x_0| < \delta_{\epsilon},</math> since <math>\left(x_n\right)</math> converges at <math>x_0</math>; combining this with <math>(*)</math> we obtain <math display=block>\forall \epsilon > 0 \,\exists \nu_{\epsilon} > 0 : \forall n > \nu_{\epsilon} \quad |f(x_n)-f(x_0)| < \epsilon.</math> Assume on the contrary that <math>f</math> is sequentially continuous and proceed by contradiction: suppose <math>f</math> is not continuous at <math>x_0</math> <math display=block>\exists \epsilon > 0 : \forall \delta_{\epsilon} > 0,\,\exists x_{\delta_{\epsilon}}: 0 < |x_{\delta_{\epsilon}}-x_0| < \delta_\epsilon \implies |f(x_{\delta_{\epsilon}})-f(x_0)| > \epsilon</math> then we can take <math>\delta_{\epsilon}=1/n,\,\forall n > 0</math> and call the corresponding point <math>x_{\delta_{\epsilon}} =: x_n</math>: in this way we have defined a sequence <math>(x_n)_{n\geq1}</math> such that <math display=block>\forall n > 0 \quad |x_n-x_0| < \frac{1}{n},\quad |f(x_n)-f(x_0)| > \epsilon</math> by construction <math>x_n \to x_0</math> but <math>f(x_n) \not\to f(x_0)</math>, which contradicts the hypothesis of sequentially continuity. <math>\blacksquare</math> Шаблон:Collapse bottom
Closure operator and interior operator definitions
In terms of the interior operator, a function <math>f : X \to Y</math> between topological spaces is continuous if and only if for every subset <math>B \subseteq Y,</math> <math display=block>f^{-1}\left(\operatorname{int}_Y B\right) ~\subseteq~ \operatorname{int}_X\left(f^{-1}(B)\right).</math>
In terms of the closure operator, <math>f : X \to Y</math> is continuous if and only if for every subset <math>A \subseteq X,</math> <math display=block>f\left(\operatorname{cl}_X A\right) ~\subseteq~ \operatorname{cl}_Y (f(A)).</math> That is to say, given any element <math>x \in X</math> that belongs to the closure of a subset <math>A \subseteq X,</math> <math>f(x)</math> necessarily belongs to the closure of <math>f(A)</math> in <math>Y.</math> If we declare that a point <math>x</math> is Шаблон:Em a subset <math>A \subseteq X</math> if <math>x \in \operatorname{cl}_X A,</math> then this terminology allows for a plain English description of continuity: <math>f</math> is continuous if and only if for every subset <math>A \subseteq X,</math> <math>f</math> maps points that are close to <math>A</math> to points that are close to <math>f(A).</math> Similarly, <math>f</math> is continuous at a fixed given point <math>x \in X</math> if and only if whenever <math>x</math> is close to a subset <math>A \subseteq X,</math> then <math>f(x)</math> is close to <math>f(A).</math>
Instead of specifying topological spaces by their open subsets, any topology on <math>X</math> can alternatively be determined by a closure operator or by an interior operator. Specifically, the map that sends a subset <math>A</math> of a topological space <math>X</math> to its topological closure <math>\operatorname{cl}_X A</math> satisfies the Kuratowski closure axioms. Conversely, for any closure operator <math>A \mapsto \operatorname{cl} A</math> there exists a unique topology <math>\tau</math> on <math>X</math> (specifically, <math>\tau := \{ X \setminus \operatorname{cl} A : A \subseteq X \}</math>) such that for every subset <math>A \subseteq X,</math> <math>\operatorname{cl} A</math> is equal to the topological closure <math>\operatorname{cl}_{(X, \tau)} A</math> of <math>A</math> in <math>(X, \tau).</math> If the sets <math>X</math> and <math>Y</math> are each associated with closure operators (both denoted by <math>\operatorname{cl}</math>) then a map <math>f : X \to Y</math> is continuous if and only if <math>f(\operatorname{cl} A) \subseteq \operatorname{cl} (f(A))</math> for every subset <math>A \subseteq X.</math>
Similarly, the map that sends a subset <math>A</math> of <math>X</math> to its topological interior <math>\operatorname{int}_X A</math> defines an interior operator. Conversely, any interior operator <math>A \mapsto \operatorname{int} A</math> induces a unique topology <math>\tau</math> on <math>X</math> (specifically, <math>\tau := \{ \operatorname{int} A : A \subseteq X \}</math>) such that for every <math>A \subseteq X,</math> <math>\operatorname{int} A</math> is equal to the topological interior <math>\operatorname{int}_{(X, \tau)} A</math> of <math>A</math> in <math>(X, \tau).</math> If the sets <math>X</math> and <math>Y</math> are each associated with interior operators (both denoted by <math>\operatorname{int}</math>) then a map <math>f : X \to Y</math> is continuous if and only if <math>f^{-1}(\operatorname{int} B) \subseteq \operatorname{int}\left(f^{-1}(B)\right)</math> for every subset <math>B \subseteq Y.</math>[16]
Filters and prefilters
Continuity can also be characterized in terms of filters. A function <math>f : X \to Y</math> is continuous if and only if whenever a filter <math>\mathcal{B}</math> on <math>X</math> converges in <math>X</math> to a point <math>x \in X,</math> then the prefilter <math>f(\mathcal{B})</math> converges in <math>Y</math> to <math>f(x).</math> This characterization remains true if the word "filter" is replaced by "prefilter."Шаблон:Sfn
Properties
If <math>f : X \to Y</math> and <math>g : Y \to Z</math> are continuous, then so is the composition <math>g \circ f : X \to Z.</math> If <math>f : X \to Y</math> is continuous and
- X is compact, then f(X) is compact.
- X is connected, then f(X) is connected.
- X is path-connected, then f(X) is path-connected.
- X is Lindelöf, then f(X) is Lindelöf.
- X is separable, then f(X) is separable.
The possible topologies on a fixed set X are partially ordered: a topology <math>\tau_1</math> is said to be coarser than another topology <math>\tau_2</math> (notation: <math>\tau_1 \subseteq \tau_2</math>) if every open subset with respect to <math>\tau_1</math> is also open with respect to <math>\tau_2.</math> Then, the identity map <math display="block">\operatorname{id}_X : \left(X, \tau_2\right) \to \left(X, \tau_1\right)</math> is continuous if and only if <math>\tau_1 \subseteq \tau_2</math> (see also comparison of topologies). More generally, a continuous function <math display="block">\left(X, \tau_X\right) \to \left(Y, \tau_Y\right)</math> stays continuous if the topology <math>\tau_Y</math> is replaced by a coarser topology and/or <math>\tau_X</math> is replaced by a finer topology.
Homeomorphisms
Symmetric to the concept of a continuous map is an open map, for which Шаблон:Em of open sets are open. If an open map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a bijective function f between two topological spaces, the inverse function <math>f^{-1}</math> need not be continuous. A bijective continuous function with a continuous inverse function is called a Шаблон:Em.
If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.
Defining topologies via continuous functions
Given a function <math display="block">f : X \to S,</math> where X is a topological space and S is a set (without a specified topology), the final topology on S is defined by letting the open sets of S be those subsets A of S for which <math>f^{-1}(A)</math> is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on S. Thus, the final topology is the finest topology on S that makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f.
Dually, for a function f from a set S to a topological space X, the initial topology on S is defined by designating as an open set every subset A of S such that <math>A = f^{-1}(U)</math> for some open subset U of X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus, the initial topology is the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X.
A topology on a set S is uniquely determined by the class of all continuous functions <math>S \to X</math> into all topological spaces X. Dually, a similar idea can be applied to maps <math>X \to S.</math>
Related notions
If <math>f : S \to Y</math> is a continuous function from some subset <math>S</math> of a topological space <math>X</math> then a Шаблон:Em of <math>f</math> to <math>X</math> is any continuous function <math>F : X \to Y</math> such that <math>F(s) = f(s)</math> for every <math>s \in S,</math> which is a condition that often written as <math>f = F\big\vert_S.</math> In words, it is any continuous function <math>F : X \to Y</math> that restricts to <math>f</math> on <math>S.</math> This notion is used, for example, in the Tietze extension theorem and the Hahn–Banach theorem. If <math>f : S \to Y</math> is not continuous, then it could not possibly have a continuous extension. If <math>Y</math> is a Hausdorff space and <math>S</math> is a dense subset of <math>X</math> then a continuous extension of <math>f : S \to Y</math> to <math>X,</math> if one exists, will be unique. The Blumberg theorem states that if <math>f : \R \to \R</math> is an arbitrary function then there exists a dense subset <math>D</math> of <math>\R</math> such that the restriction <math>f\big\vert_D : D \to \R</math> is continuous; in other words, every function <math>\R \to \R</math> can be restricted to some dense subset on which it is continuous.
Various other mathematical domains use the concept of continuity in different but related meanings. For example, in order theory, an order-preserving function <math>f : X \to Y</math> between particular types of partially ordered sets <math>X</math> and <math>Y</math> is continuous if for each directed subset <math>A</math> of <math>X,</math> we have <math>\sup f(A) = f(\sup A).</math> Here <math>\,\sup\,</math> is the supremum with respect to the orderings in <math>X</math> and <math>Y,</math> respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the Scott topology.[17][18]
In category theory, a functor <math display="block">F : \mathcal C \to \mathcal D</math> between two categories is called Шаблон:Em if it commutes with small limits. That is to say, <math display="block">\varprojlim_{i \in I} F(C_i) \cong F \left(\varprojlim_{i \in I} C_i \right)</math> for any small (that is, indexed by a set <math>I,</math> as opposed to a class) diagram of objects in <math>\mathcal C</math>.
A Шаблон:Em is a generalization of metric spaces and posets,[19][20] which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains.[21]
See also
- Continuity (mathematics)
- Absolute continuity
- Dini continuity
- Equicontinuity
- Geometric continuity
- Parametric continuity
- Classification of discontinuities
- Coarse function
- Continuous function (set theory)
- Continuous stochastic process
- Normal function
- Open and closed maps
- Piecewise
- Symmetrically continuous function
- Direction-preserving function - an analog of a continuous function in discrete spaces.
References
Шаблон:Commons category Шаблон:Reflist
Bibliography
Шаблон:Calculus topics Шаблон:Analysis-footer Шаблон:Authority control
- ↑ Шаблон:Cite web
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Шаблон:Citation, section II.4
- ↑ Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172
- ↑ Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177
- ↑ Шаблон:Citation
- ↑ Шаблон:Citation, section IV.10
- ↑ Шаблон:Citation, section 9.4
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
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