Uniform convergence

A sequence of functions is said to uniformly converge to a function if for every positive real number , then there exists such that for all positive integers , we have .

Uniformly convergent sequences have a number of nice properties that pointwise convergent sequences do not necessarily have. A sequence of continuous uniformly convergent functions converge to a continuous function. A sequence of differentiable uniformly convergent functions (on a closed interval) converge to a differentiable function, and a sequence of Stieltjes-integrable functions converge to a Stieltjes-integrable function.

This article is a stub. Help us out by expanding it.