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Dini continuity : ウィキペディア英語版
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^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(f(x),f(y)). \,
The function f is called Dini-continuous if
:\int_0^1 \frac\,dt < \infty.
An equivalent condition is that, for any \theta \in (0,1),
:\sum_^\infty \omega_f(\theta^i a) < \infty
where a is the diameter of X.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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