Dini continuity explained

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 R ^{n} ), and let f:X → X be a function from X into itself. The modulus of continuity of f is\omega _{f(t)} = \sup _{d(x,y)\le} d(f(x),f(y)).

The function

f is called Dini-continuous if\int ^{1} _{0} (\omega _{f(t)})/(t) dt < inf ty.

An equivalent condition is that, for any

\theta \in ( 0 , 1 ) ,\sum ^{inf ty} _{i =1} \omega ^{i} _{f(\theta} a) < inf ty

where

a is the diameter of X .

See also

• Dini test — a condition similar to local Dini continuity implies convergence of a Fourier transform.

References

• Örjan . Stenflo . A note on a theorem of Karlin . Statistics & Probability Letters . 54 . 2 . 2001 . 183–187 . 10.1016/S0167-7152(01)00045-1 .