Editing Equicontinuity (section)

==Equicontinuity and uniform convergence==

Let ''X'' be a compact Hausdorff space, and equip ''C''(''X'') with the [[uniform norm]], thus making ''C''(''X'') a [[Banach space]], hence a metric space.  Then [[Arzelà–Ascoli theorem]] states that a subset of ''C''(''X'') is compact if and only if it is closed, uniformly bounded and equicontinuous. {{sfn|Rudin|1991|p=394 Appendix A5}}
This is analogous to the [[Heine–Borel theorem]], which states that subsets of '''R'''<sup>''n''</sup> are compact if and only if they are closed and bounded.{{sfn|Rudin|1991|p=18 Theorem 1.23}} 
As a corollary, every uniformly bounded equicontinuous sequence in ''C''(''X'') contains a subsequence that converges uniformly to a continuous function on ''X''.

In view of Arzelà–Ascoli theorem, a sequence in ''C''(''X'') converges uniformly if and only if it is equicontinuous and converges pointwise. The hypothesis of the statement can be weakened a bit: a sequence in ''C''(''X'') converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function on ''X''  (not assumed continuous).
{{Math proof|drop=hidden|proof=
Suppose ''f''<sub>''j''</sub> is an equicontinuous sequence of continuous functions on a dense subset ''D'' of ''X''. 
Let ''ε''&nbsp;>&nbsp;0 be given. By equicontinuity, for each {{nowrap|''z'' ∈ ''D''}}, there exists a neighborhood ''U''<sub>z</sub> of ''z'' such that
: <math>|f_j(x) - f_j(z)| < \epsilon / 3 </math>
for all ''j'' and {{nowrap|''x'' ∈ ''U<sub>z</sub>''}}. 
By denseness and compactness, we can find a finite subset {{nowrap|''D′'' ⊂ ''D''}} such that ''X'' is the union of ''U<sub>z</sub>'' over {{nowrap|''z'' ∈ ''D′''}}. Since ''f''<sub>''j''</sub> converges pointwise on {{nowrap|''D′''}}, there exists ''N'' > 0 such that
: <math>|f_j(z) - f_k(z)| < \epsilon / 3 </math>
whenever {{nowrap|''z'' ∈ ''D′''}} and ''j'', ''k'' > ''N''. It follows that
: <math>\sup_X |f_j - f_k| < \epsilon</math>
for all ''j'', ''k'' > ''N''. In fact, if {{nowrap|''x'' ∈ ''X''}}, then {{nowrap|''x'' ∈ ''U<sub>z</sub>''}} for some {{nowrap|''z'' ∈ ''D′''}} and so we get:
: <math>|f_j(x) - f_k(x)| \le |f_j(x) - f_j(z)| + |f_j(z) - f_k(z)| + |f_k(z) - f_k(x)| < \epsilon</math>.
Hence, ''f''<sub>''j''</sub> is Cauchy in ''C''(''X'') and thus converges by completeness.
}}

This weaker version is typically used to prove Arzelà–Ascoli theorem for separable compact spaces. Another consequence is that the limit of an equicontinuous pointwise convergent sequence of continuous functions on a metric space, or on a locally compact space, is continuous. (See below for an example.) 
In the above, the hypothesis of compactness of ''X''&thinsp; cannot be relaxed. 
To see that, consider a compactly supported continuous function ''g'' on '''R''' with ''g''(0)&nbsp;= 1, and consider the equicontinuous sequence of functions {{mset|''ƒ''<sub>''n''</sub>}} on '''R''' defined by ''ƒ''<sub>''n''</sub>(''x'')&nbsp;= {{nowrap|''g''(''x'' − ''n'')}}.  Then, ''ƒ''<sub>''n''</sub> converges pointwise to 0 but does not converge uniformly to 0.

This criterion for uniform convergence is often useful in real and complex analysis. Suppose we are given a sequence of continuous functions that converges pointwise on some open subset ''G'' of '''R'''<sup>''n''</sup>. As noted above, it actually converges uniformly on a compact subset of ''G'' if it is equicontinuous on the compact set. In practice, showing the equicontinuity is often not so difficult. For example, if the sequence consists of differentiable functions or functions with some regularity (e.g., the functions are solutions of a differential equation), then the [[mean value theorem]] or some other kinds of estimates can be used to show the sequence is equicontinuous. It then follows that the limit of the sequence is continuous on every compact subset of ''G''; thus, continuous on ''G''. A similar argument can be made when the functions are holomorphic. One can use, for instance, [[Cauchy's estimate]] to show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic. Note that the equicontinuity is essential here. For example, ''ƒ''<sub>''n''</sub>(''x'')&nbsp;= {{nowrap|arctan ''n''&thinsp;''x''}} converges to a multiple of the discontinuous [[sign function]].