Dini continuity - Alchetron, The Free Social Encyclopedia

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

ω f ( t ) = sup d ( x , y ) ≤ t d ( f ( x ) , f ( y ) ) .

The function f is called Dini-continuous if

∫ 0 1 ω f ( t ) t d t < ∞ .

An equivalent condition is that, for any θ ∈ ( 0 , 1 ) ,

∑ i = 1 ∞ ω f ( θ i a ) < ∞

where a is the diameter of X .

References

Dini continuity Wikipedia

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