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 $$X$$.