# About

Tag Info

For sequences of functions, uniform convergence is a mode of convergence stronger than pointwise convergence, preserving certain properties such as continuity. This tag should be used with the tag [convergence].

A sequence of functions $f_n : X \to \mathbb R$ is said to converge uniformly to a function $f : X \to \mathbb R$ if $$\lim_{n\to \infty}\sup_{x\in X}|f_n(x)-f(x)| = 0.$$ Roughly speaking, this means not only that $f_n(x)$ converges to $f(x)$ for all $x \in X$, but also that the rate of convergence is uniform over the whole of $X$.

Uniformly convergent sequences are well-behaved in certain ways that pointwise convergent sequences are not. For example, if $X$ is a topological space (such as a subset of $\mathbb R$), and if the functions $\{ f_n \}$ are continuous, then their uniform limit $f$ is also continuous. Furthermore, if $X$ is a bounded closed interval in $\mathbb R$, and if the functions $\{ f_n \}$ are Riemann integrable, then their uniform limit $f$ is also Riemann integrable, and $\int_X f= \lim_{n \to \infty} \int_X f_n$. These statements do not hold under the weaker assumption of pointwise convergence.