Dini Continuity

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 $X$ be a compact subset of a metric space (such as $\mathbb^n$), and let $f:X\rightarrow X$ be a function from $X$ into itself. The modulus of continuity of $f$ is

:$\omega_f(t) = \sup_ d(f(x),f(y)).$

The function $f$ is called Dini-continuous if

:$\int_0^1 \frac\,dt < \infty.$

An equivalent condition is that, for any $\theta \in (0,1)$,

:$\sum_^\infty \omega_f(\theta^i a) < \infty$

where $a$ is the diameter of $X$.

See also

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

References

*

{{Mathanalysis-stub

Mathematical analysis