Dini continuity

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

:$\backslash omega\_f(t)\; =\; \backslash sup\_\{d(x,y)\backslash le\; t\}\; d(f(x),f(y)).$

The function $f$ is called Dini-continuous if

:

An equivalent condition is that, for any $\backslash theta\; \backslash in\; (0,1)$,

:

where $a$ is the diameter of $X$.

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

• {{cite journal |first=Örjan |last=Stenflo |title=A note on a theorem of Karlin |journal=Statistics & Probability Letters |volume=54 |issue=2 |year=2001 |pages=183–187 |doi=10.1016/S0167-7152(01)00045-1 }}

{{Mathanalysis-stub}}

Category:Mathematical analysis