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 $\mathbb{R}^n$ ), 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(x,y)\le t} d(f(x),f(y)). \,

The function $f$ is called Dini-continuous if

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

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

\sum_{i=1}^\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

Stenflo, Örjan (2001). "A note on a theorem of Karlin". Statistics & Probability Letters 54 (2): 183–187. doi:10.1016/S0167-7152(01)00045-1.