- Dini continuity
-
In mathematical analysis, Dini continuity is a refinement of continuity.
Contents
Definition
Let X be a compact subset of a metric space (such as ), and let be a function from X into itself. The modulus of continuity of f is
The function f is called Dini-continuous if
An equivalent condition is that, for any ,
where a is the diameter of X.
Properties
If f is Dini continuous then it is continuous.
If f is Lipschitz continuous then it is Dini continuous.
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.
Categories:
Wikimedia Foundation. 2010.