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 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
- 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.
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