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

[math]\displaystyle{ \omega_f(t) = \sup_{d(x,y)\le t} d(f(x),f(y)). }[/math]

The function [math]\displaystyle{ f }[/math] is called Dini-continuous if

[math]\displaystyle{ \int_0^1 \frac{\omega_f(t)}{t}\,dt \lt \infty. }[/math]

An equivalent condition is that, for any [math]\displaystyle{ \theta \in (0,1) }[/math],

[math]\displaystyle{ \sum_{i=1}^\infty \omega_f(\theta^i a) \lt \infty }[/math]

where [math]\displaystyle{ a }[/math] is the diameter of [math]\displaystyle{ X }[/math].

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.