Английская Википедия:Dini continuity

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In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous.

Definition

Let <math>X</math> be a compact subset of a metric space (such as <math>\mathbb{R}^n</math>), and let <math>f:X\rightarrow X</math> be a function from <math>X</math> into itself. The modulus of continuity of <math>f</math> is

<math>\omega_f(t) = \sup_{d(x,y)\le t} d(f(x),f(y)). </math>

The function <math>f</math> is called Dini-continuous if

<math>\int_0^1 \frac{\omega_f(t)}{t}\,dt < \infty.</math>

An equivalent condition is that, for any <math>\theta \in (0,1)</math>,

<math>\sum_{i=1}^\infty \omega_f(\theta^i a) < \infty</math>

where <math>a</math> is the diameter of <math>X</math>.

See also

• Dini test — a condition similar to local Dini continuity implies convergence of a Fourier transform.

References

• Шаблон:Cite journal