Dini continuity

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

Definition

Let X {\displaystyle X} {\displaystyle X} be a compact subset of a metric space (such as R n {\displaystyle \mathbb {R} ^{n}} {\displaystyle \mathbb {R} ^{n}}), and let f : X → X {\displaystyle f:X\rightarrow X} {\displaystyle f:X\rightarrow X} be a function from X {\displaystyle X} {\displaystyle X} into itself. The modulus of continuity of f {\displaystyle f} {\displaystyle f} is

ω f ( t ) = sup d ( x , y ) ≤ t d ( f ( x ) , f ( y ) ) . {\displaystyle \omega _{f}(t)=\sup _{d(x,y)\leq t}d(f(x),f(y)).} {\displaystyle \omega _{f}(t)=\sup _{d(x,y)\leq t}d(f(x),f(y)).}

The function f {\displaystyle f} {\displaystyle f} is called Dini-continuous if

∫ 0 1 ω f ( t ) t d t < ∞ . {\displaystyle \int _{0}^{1}{\frac {\omega _{f}(t)}{t}}\,dt<\infty .} {\displaystyle \int _{0}^{1}{\frac {\omega _{f}(t)}{t}}\,dt<\infty .}

An equivalent condition is that, for any θ ∈ ( 0 , 1 ) {\displaystyle \theta \in (0,1)} {\displaystyle \theta \in (0,1)},

∑ i = 1 ∞ ω f ( θ i a ) < ∞ {\displaystyle \sum _{i=1}^{\infty }\omega _{f}(\theta ^{i}a)<\infty } {\displaystyle \sum _{i=1}^{\infty }\omega _{f}(\theta ^{i}a)<\infty }

where a {\displaystyle a} {\displaystyle a} is the diameter of X {\displaystyle X} {\displaystyle X}.

See also

References