Dini continuity

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


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.

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