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

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

The function ${\displaystyle f}$ is called Dini-continuous if

${\displaystyle \int _{0}^{1}{\frac {\omega _{f}(t)}{t}}\,dt<\infty .}$

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

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

where ${\displaystyle a}$ is the diameter of ${\displaystyle X}$.

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.