Uniform convergence

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

Every uniformly convergent sequence converges pointwise, but the converse is not necessarily true. For example, the sequence of functions defined by $f_n(x) = x^n$ for $x \in [0, 1]$ converges pointwise to the function $f(x) =\begin{cases} 1, & x = 1 \\0, & \text{otherwise}\end{cases}$ , but this convergence is not uniform.

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

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Uniform convergence