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.

Definition

Let be a compact subset of a metric space (such as ), and let be a function from into itself. The modulus of continuity of is

The function is called Dini-continuous if

An equivalent condition is that, for any ,

where is the diameter of .

See also

References

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