# 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 $(f_{n})$ converges uniformly to a limiting function $f$ on a set $E$ as the function domain if, given any arbitrarily small positive number $\epsilon$, a number $N$ can be found such that each of the functions $f_{N},f_{N+1},f_{N+2},\ldots$ differs from $f$ by no more than $\epsilon$ at every point $x$ in $E$. Described in an informal way, if $f_{n}$ converges to $f$ uniformly, then how quickly the functions $f_{n}$ approach $f$ is "uniform" throughout $E$ in the following sense: in order to guarantee that $f_{n}(x)$ differs from $f(x)$ by less than a chosen distance $\epsilon$, we only need to make sure that $n$ is larger than or equal to a certain $N$, which we can find without knowing the value of $x\in E$ in advance. In other words, there exists a number $N=N(\epsilon )$ that could depend on $\epsilon$ but is independent of $x$, such that choosing $n\geq N$ will ensure that $|f_{n}(x)-f(x)|<\epsilon$ for all $x\in E$. In contrast, pointwise convergence of $f_{n}$ to $f$ merely guarantees that for any $x\in E$ given in advance, we can find $N=N(\epsilon ,x)$ (i.e., $N$ could depend on the values of both $\epsilon$ and $x$) such that, for that particular $x$, $f_{n}(x)$ falls within $\epsilon$ of $f(x)$ whenever $n\geq N$ (and a different $x$ may require a different, larger $N$ for $n\geq N$ to guarantee that $|f_{n}(x)-f(x)|<\epsilon$).
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 $f_{n}$, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit $f$ if the convergence is uniform, but not necessarily if the convergence is not uniform.

## History

In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to which Niels Henrik Abel in 1826 found purported counterexamples in the context of Fourier series, arguing that Cauchy's proof had to be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions.
The term uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series ∑$_{n=1}$$^∞$f$_n$(x, ϕ, ψ){\textstyle \sum _{n=1}^{\infty }f_{n}(x, \phi , \psi )} is independent of the variables $\phi$ and $\psi .$ While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs.
Later Gudermann's pupil Karl Weierstrass, who attended his course on elliptic functions in 1839–1840, coined the term gleichmäßig konvergent (German: uniformly convergent) which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894. Independently, similar concepts were articulated by Philipp Ludwig von Seidel and George Gabriel Stokes. G. H. Hardy compares the three definitions in his paper "Sir George Stokes and the concept of uniform convergence" and remarks: "Weierstrass's discovery was the earliest, and he alone fully realized its far-reaching importance as one of the fundamental ideas of analysis."
Under the influence of Weierstrass and Bernhard Riemann this concept and related questions were intensely studied at the end of the 19th century by Hermann Hankel, Paul du Bois-Reymond, Ulisse Dini, Cesare Arzelà and others.

## Definition

We first define uniform convergence for real-valued functions, although the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces (see below).
Suppose $E$ is a set and $(f_{n})_{n\in \mathbb {N} }$ is a sequence of real-valued functions on it. We say the sequence $(f_{n})_{n\in \mathbb {N} }$ is uniformly convergent on $E$ with limit $f:E\to \mathbb {R}$ if for every $\epsilon >0,$ there exists a natural number $N$ such that for all $n\geq N$ and for all $x\in E$
    $|f_{n}(x)-f(x)|<\epsilon .$
The notation for uniform convergence of $f_{n}$ to $f$ is not quite standardized and different authors have used a variety of symbols, including (in roughly decreasing order of popularity):
    $f_{n}\rightrightarrows f,\quad {\underset {n\to \infty }{\mathrm {unif\ lim} }}f_{n}=f,\quad f_{n}{\overset {\mathrm {unif.} }{\longrightarrow }}f,\quad f=\mathrm {u} \!\!-\!\!\!\lim _{n\to \infty }f_{n}.$
Frequently, no special symbol is used, and authors simply write
    $f_{n}\to f\quad \mathrm {uniformly}$
to indicate that convergence is uniform. (In contrast, the expression $f_{n}\to f$ on $E$ without an adverb is taken to mean pointwise convergence on $E$: for all $x\in E$, $f_{n}(x)\to f(x)$ as $n\to \infty$.)
Since $\mathbb {R}$ is a complete metric space, the Cauchy criterion can be used to give an equivalent alternative formulation for uniform convergence: $(f_{n})_{n\in \mathbb {N} }$ converges uniformly on $E$ (in the previous sense) if and only if for every $\epsilon >0$, there exists a natural number $N$ such that
    $x\in E,m,n\geq N\implies |f_{m}(x)-f_{n}(x)|<\epsilon$.
In yet another equivalent formulation, if we define
    $d_{n}=\sup _{x\in E}|f_{n}(x)-f(x)|,$
then $f_{n}$ converges to $f$ uniformly if and only if $d_{n}\to 0$ as $n\to \infty$. Thus, we can characterize uniform convergence of $(f_{n})_{n\in \mathbb {N} }$ on $E$ as (simple) convergence of $(f_{n})_{n\in \mathbb {N} }$ in the function space $\mathbb {R} ^{E}$ with respect to the uniform metric (also called the supremum metric), defined by
    $d(f,g)=\sup _{x\in E}|f(x)-g(x)|.$
Symbolically,
    $f_{n}\rightrightarrows f\iff d(f_{n},f)\to 0$.
The sequence $(f_{n})_{n\in \mathbb {N} }$ is said to be locally uniformly convergent with limit $f$ if $E$ is a metric space and for every $x\in E$, there exists an $r>0$ such that $(f_{n})$ converges uniformly on $B(x,r)\cap E.$ It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence.