Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions ${\displaystyle (f_{n})}$ converges uniformly to a limiting function ${\displaystyle f}$ on a set ${\displaystyle E}$ as the function domain if, given any arbitrarily small positive number ${\displaystyle \epsilon }$, a number ${\displaystyle N}$ can be found such that each of the functions ${\displaystyle f_{N},f_{N+1},f_{N+2},\ldots }$ differs from ${\displaystyle f}$ by no more than ${\displaystyle \epsilon }$ at every point ${\displaystyle x}$ in ${\displaystyle E}$. Described in an informal way, if ${\displaystyle f_{n}}$ converges to ${\displaystyle f}$ uniformly, then how quickly the functions ${\displaystyle f_{n}}$ approach ${\displaystyle f}$ is "uniform" throughout ${\displaystyle E}$ in the following sense: in order to guarantee that ${\displaystyle f_{n}(x)}$ differs from ${\displaystyle f(x)}$ by less than a chosen distance ${\displaystyle \epsilon }$, we only need to make sure that ${\displaystyle n}$ is larger than or equal to a certain ${\displaystyle N}$, which we can find without knowing the value of ${\displaystyle x\in E}$ in advance. In other words, there exists a number ${\displaystyle N=N(\epsilon )}$ that could depend on ${\displaystyle \epsilon }$ but is independent of ${\displaystyle x}$, such that choosing ${\displaystyle n\geq N}$ will ensure that ${\displaystyle |f_{n}(x)-f(x)|<\epsilon }$ for all ${\displaystyle x\in E}$. In contrast, pointwise convergence of ${\displaystyle f_{n}}$ to ${\displaystyle f}$ merely guarantees that for any ${\displaystyle x\in E}$ given in advance, we can find ${\displaystyle N=N(\epsilon ,x)}$ (i.e., ${\displaystyle N}$ could depend on the values of both ${\displaystyle \epsilon }$ and ${\displaystyle x}$) such that, for that particular ${\displaystyle x}$, ${\displaystyle f_{n}(x)}$ falls within ${\displaystyle \epsilon }$ of ${\displaystyle f(x)}$ whenever ${\displaystyle n\geq N}$ (and a different ${\displaystyle x}$ may require a different, larger ${\displaystyle N}$ for ${\displaystyle n\geq N}$ to guarantee that ${\displaystyle |f_{n}(x)-f(x)|<\epsilon }$).

The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions ${\displaystyle f_{n}}$, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit ${\displaystyle f}$ if the convergence is uniform, but not necessarily if the convergence is not uniform.