Editing Uniform convergence (section)

{{Short description|Mode of convergence of a function sequence}}
[[File:Uniform convergence.svg|thumb|300x300px|A sequence of functions <math>(f_n)</math> converges uniformly to <math>f</math> when for arbitrary small <math>\varepsilon</math> there is an index <math>N</math> such that the graph of <math>f_n</math> is in the <math>\varepsilon</math>-tube around f whenever <math>n\ge N.</math>]]
[[File:Drini-nonuniformconvergence.png|thumb|300x300px|
The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions <math>f_n(x)=\sin^n(x)</math> (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).]]
In the [[mathematics|mathematical]] field of [[mathematical analysis|analysis]], '''uniform convergence''' is a [[Modes of convergence|mode of convergence]] of functions stronger than [[pointwise convergence]]. A [[sequence]] of [[Function (mathematics)|functions]] <math>(f_n)</math> '''converges uniformly''' to a limiting function <math>f</math> on a set <math>E</math> as the function domain if, given any arbitrarily small positive number <math>\varepsilon</math>, a number <math>N</math> can be found such that each of the functions <math>f_N, f_{N+1},f_{N+2},\ldots</math> differs from <math>f</math> by no more than <math>\varepsilon</math> ''at every point'' <math>x</math> ''in'' <math>E</math>.  Described in an informal way, if <math>f_n</math> converges to <math>f</math> uniformly, then how quickly the functions <math>f_n</math> approach <math>f</math> is "uniform" throughout <math>E</math> in the following sense: in order to guarantee that <math>f_n(x)</math> differs from <math>f(x)</math> by less than a chosen distance <math>\varepsilon</math>, we only need to make sure that <math>n</math> is larger than or equal to a certain <math>N</math>, which we can find without knowing the value of <math>x\in E</math> in advance.  In other words, there exists a number <math>N=N(\varepsilon)</math> that could depend on <math>\varepsilon</math> but is ''independent of <math>x</math>'', such that choosing <math>n\geq N</math> will ensure that <math>|f_n(x)-f(x)|<\varepsilon</math> ''for all <math>x\in E</math>''. In contrast, pointwise convergence of <math>f_n</math> to <math>f</math> merely guarantees that for any <math>x\in E</math> given in advance, we can find <math>N=N(\varepsilon, x)</math> (i.e., <math>N</math> could depend on the values of both <math>\varepsilon</math> and'' <math>x</math>'') such that, ''for that particular'' ''<math>x</math>'', <math>f_n(x)</math> falls within <math>\varepsilon</math> of <math>f(x)</math> whenever <math>n\geq N</math> (and a different <math>x</math> may require a different, larger <math>N</math> for <math>n\geq N</math> to guarantee that <math>|f_n(x)-f(x)|<\varepsilon</math>).

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 <math>f_n</math>, such as [[continuous function|continuity]], [[Riemann integral|Riemann integrability]], and, with additional hypotheses, [[differentiability]], are transferred to the [[limit of a function|limit]] <math>f</math> if the convergence is uniform, but not necessarily if the convergence is not uniform.