Uniform convergence

A sequence of functions $\{f_n\},\ f_n: X \to \mathbb{R}$ is said to uniformly converge to a function $f: X \to Y$ if for every positive real number $\varepsilon > 0$ , then there exists $N$ such that for all positive integers $n \ge N$ , we have $|f_n(x) - f(x)| < \varepsilon$ .

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.

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Uniform convergence