Dini continuity

Summary

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

Definition edit

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 edit

References edit

  • 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.