Английская Википедия:Conway base 13 function
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The Conway base 13 function is a function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem. In other words, it is a function that satisfies a particular intermediate-value property—on any interval (a, b), the function f takes every value between f(a) and f(b)—but is not continuous.
In 2018, a much simpler function with the property that every open set is mapped onto the full real line was constructed by Aksel Bergfeldt.[1] This function is also nowhere continuous.
Purpose
The Conway base 13 function was created as part of a "produce" activity: in this case, the challenge was to produce a simple-to-understand function which takes on every real value in every interval, that is, it is an everywhere surjective function.[2] It is thus discontinuous at every point.
Sketch of definition
- Every real number x can be represented in base 13 in a unique canonical way; such representations use the digits 0–9 plus three additional symbols, say {A, B, C}. For example, the number 54349589 has a base-13 representation
B34C128. - If instead of {A, B, C}, we judiciously choose the symbols {+, −, .}, some numbers in base 13 will have representations that look like well-formed decimals in base 10: for example, the number 54349589 has a base-13 representation of
−34.128. Of course, most numbers will not be intelligible in this way; for example, the number 3629265 has the base-13 representation9+0−−7. - Conway's base-13 function takes in a real number x and considers its base-13 representation as a sequence of symbols Шаблон:Nowrap. If from some position onward, the representation looks like a well-formed decimal number r, then f(x) = r. Otherwise, f(x) = 0. (Well-formed means that it starts with a + or − symbol, contains exactly one decimal-point symbol, and otherwise contains only the digits 0–9). For example, if a number x has the representation
8++2.19+0−−7+3.141592653..., then f(x) = +3.141592653....
Definition
The Conway base-13 function is a function <math>f: \Reals \to \Reals</math> defined as follows. Write the argument <math>x</math> value as a tridecimal (a "decimal" in base 13) using 13 symbols as "digits": Шаблон:Nowrap; there should be no trailing C recurring. There may be a leading sign, and somewhere there will be a tridecimal point to separate the integer part from the fractional part; these should both be ignored in the sequel. These "digits" can be thought of as having the values 0 to 12 respectively; Conway originally used the digits "+", "−" and "." instead of A, B, C, and underlined all of the base-13 "digits" to clearly distinguish them from the usual base-10 digits and symbols.
- If from some point onwards, the tridecimal expansion of <math>x</math> is of the form <math>A x_1 x_2 \dots x_n C y_1 y_2 \dots</math> where all the digits <math>x_i</math> and <math>y_j</math> are in <math>\{0, \dots, 9\},</math> then <math>f(x) = x_1 \dots x_n . y_1 y_2 \dots</math> in usual base-10 notation.
- Similarly, if the tridecimal expansion of <math>x</math> ends with <math>B x_1 x_2 \dots x_n C y_1 y_2 \dots,</math> then <math>f(x) = -x_1 \dots x_n . y_1 y_2 \dots.</math>
- Otherwise, <math>f(x) = 0.</math>
For example:
- <math>f(\mathrm{12345A3C14.159} \dots_{13}) = f(\mathrm{A3C14.159} \dots_{13}) = 3.14159 \dots,</math>
- <math>f(\mathrm{B1C234}_{13}) = -1.234,</math>
- <math>f(\mathrm{1C234A567}_{13}) = 0.</math>
Properties
- According to the intermediate-value theorem, every continuous real function <math>f</math> has the intermediate-value property: on every interval (a, b), the function <math>f</math> passes through every point between <math>f(a)</math> and <math>f(b).</math> The Conway base-13 function shows that the converse is false: it satisfies the intermediate-value property, but is not continuous.
- In fact, the Conway base-13 function satisfies a much stronger intermediate-value property—on every interval (a, b), the function <math>f</math> passes through every real number. As a result, it satisfies a much stronger discontinuity property— it is discontinuous everywhere.
- From the above follows even more regarding the discontinuity of the function - its graph is dense in <math>\mathbb{R}^2</math>.
- To prove that the Conway base-13 function satisfies this stronger intermediate property, let (a, b) be an interval, let c be a point in that interval, and let r be any real number. Create a base-13 encoding of r as follows: starting with the base-10 representation of r, replace the decimal point with C and indicate the sign of r by prepending either an A (if r is positive) or a B (if r is negative) to the beginning. By definition of the Conway base-13 function, the resulting string <math>\hat{r}</math> has the property that <math>f(\hat{r}) = r.</math> Moreover, any base-13 string that ends in <math>\hat{r}</math> will have this property. Thus, if we replace the tail end of c with <math>\hat{r},</math> the resulting number will have f(cШаблон:') = r. By introducing this modification sufficiently far along the tridecimal representation of <math>c,</math> you can ensure that the new number <math>c'</math> will still lie in the interval <math>(a, b).</math> This proves that for any number r, in every interval we can find a point <math>c'</math> such that <math>f(c') = r.</math>
- The Conway base-13 function is therefore discontinuous everywhere: a real function that is continuous at x must be locally bounded at x, i.e. it must be bounded on some interval around x. But as shown above, the Conway base-13 function is unbounded on every interval around every point; therefore it is not continuous anywhere.
- The Conway base-13 function maps almost all the reals in any interval to 0.[3]
See also
References
