Editing Real analysis (section)

==== Uniform and pointwise convergence for sequences of functions ====
{{Main|Uniform convergence}}
In addition to sequences of numbers, one may also speak of ''sequences of functions'' ''on'' <math>E\subset \mathbb{R}</math>, that is, infinite, ordered families of functions <math>f_n:E\to\mathbb{R}</math>, denoted <math>(f_n)_{n=1}^\infty</math>, and their convergence properties.  However, in the case of sequences of functions, there are two kinds of convergence, known as ''pointwise convergence'' and ''uniform convergence'', that need to be distinguished.

Roughly speaking, pointwise convergence of functions <math>f_n</math> to a limiting function <math>f:E\to\mathbb{R}</math>, denoted <math>f_n \rightarrow f</math>, simply means that given any <math>x\in E</math>, <math>f_n(x)\to f(x)</math> as <math>n\to\infty</math>.  In contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely.  [[Uniform convergence]] requires members of the family of functions, <math>f_n</math>, to fall within some error <math>\varepsilon > 0</math> of <math>f</math> for ''every value of <math>x\in E</math>'', whenever <math>n\geq N</math>, for some integer <math>N</math>.  For a family of functions to uniformly converge, sometimes denoted <math>f_n\rightrightarrows f</math>, such a value of <math>N</math> must exist for any <math>\varepsilon>0</math> given, no matter how small.  Intuitively, we can visualize this situation by imagining that, for a large enough <math>N</math>, the functions <math>f_N, f_{N+1}, f_{N+2},\ldots</math> are all confined within a 'tube' of width <math>2\varepsilon</math> about <math>f</math> (that is, between <math>f - \varepsilon</math> and <math>f+\varepsilon</math>) ''for every value in their domain'' <math>E</math>.

The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence.  For example, a sequence of continuous functions (see [[Real analysis#Continuity|below]]) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise.  [[Karl Weierstrass]] is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.