Dini continuity

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

$\omega _{f}(t)=\sup _{d(x,y)\leq t}d(f(x),f(y)).$

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.