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 ${ℝ}^n$), and let $f:X\to X$ be a function from $X$ into itself. The modulus of continuity of $f$ is

${\omega }_f(t)=\underset{d(x,y)\le t}{\sup }d(f(x),f(y)).$

The function $f$ is called Dini-continuous if

$\underset{0}{\overset{1}{\int }}\frac{{\omega }_f(t)}{t}dt<\mathrm{\infty }.$

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

$\sum _{i=1}^{\mathrm{\infty }}{\omega }_f({\theta }^{i}a)<\mathrm{\infty }$

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

See also

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