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} be a compact subset of a metric space (such as R n {\displaystyle \mathbb {R} ^{n}} ), and let f : X → X {\displaystyle f:X\rightarrow X} be a function from X {\displaystyle X} into itself. The modulus of continuity of 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)).}
The function 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 .}
An equivalent condition is that, for any θ ∈ ( 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 }
where a {\displaystyle a} is the diameter of X {\displaystyle X} .
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.