Editing Equicontinuity (section)

== Equicontinuity between metric spaces ==

Let ''X'' and ''Y'' be two [[metric space]]s, and ''F'' a family of functions from ''X'' to ''Y''. We shall denote by ''d'' the respective metrics of these spaces.

The family&nbsp;''F'' is '''equicontinuous at a point''' ''x''<sub>0</sub>&nbsp;∈&nbsp;''X'' if for every ε&nbsp;>&nbsp;0, there exists a&nbsp;δ&nbsp;>&nbsp;0 such that ''d''(''ƒ''(''x''<sub>0</sub>),&nbsp;''ƒ''(''x''))&nbsp;<&nbsp;ε for all&nbsp;''ƒ''&nbsp;∈&nbsp;''F'' and all ''x'' such that ''d''(''x''<sub>0</sub>,&nbsp;''x'')&nbsp;<&nbsp;δ. 
The family is '''pointwise equicontinuous''' if it is equicontinuous at each point of ''X''.<ref name=RS29>{{harvtxt|Reed|Simon|1980}}, p. 29; {{harvtxt|Rudin|1987}}, p. 245</ref>

The family&nbsp;''F'' is '''uniformly equicontinuous''' if for every ε&nbsp;>&nbsp;0, there exists a&nbsp;δ&nbsp;>&nbsp;0 such that  ''d''(''ƒ''(''x''<sub>1</sub>),&nbsp;''ƒ''(''x''<sub>2</sub>))&nbsp;<&nbsp;ε for all ''ƒ''&nbsp;∈&nbsp;''F'' and all ''x''<sub>1</sub>, ''x''<sub>2</sub>&nbsp;∈&nbsp;''X'' such that ''d''(''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>)&nbsp;<&nbsp;δ.<ref>{{harvtxt|Reed|Simon|1980}}, p. 29</ref>

For comparison, the statement 'all functions ''ƒ'' in ''F'' are continuous' means that for every ε&nbsp;>&nbsp;0, every&nbsp;''ƒ''&nbsp;∈&nbsp;''F'', and every ''x''<sub>0</sub>&nbsp;∈&nbsp;''X'', there exists a&nbsp;δ&nbsp;>&nbsp;0 such that ''d''(''ƒ''(''x''<sub>0</sub>),&nbsp;''ƒ''(''x''))&nbsp;<&nbsp;ε for all ''x''&nbsp;∈&nbsp;''X'' such that ''d''(''x''<sub>0</sub>,&nbsp;''x'')&nbsp;<&nbsp;δ.

* For ''[[Continuous function|continuity]]'', δ&nbsp;may depend on&nbsp;ε, ''ƒ'', and ''x''<sub>0</sub>.
* For ''[[uniform continuity]]'', δ&nbsp;may depend on&nbsp;ε and&nbsp;''ƒ''.
* For ''pointwise equicontinuity'', δ&nbsp;may depend on&nbsp;ε and ''x''<sub>0</sub>.
* For ''uniform equicontinuity'', δ&nbsp;may depend only on&nbsp;ε.

More generally, when ''X'' is a topological space, a set ''F'' of functions from ''X'' to ''Y'' is said to be equicontinuous at ''x'' if for every ε&nbsp;>&nbsp;0,  ''x'' has a neighborhood ''U<sub>x</sub>'' such that
: <math>d_Y(f(y), f(x)) < \epsilon </math>
for all {{nowrap|''y'' ∈ ''U<sub>x</sub>''}} and ''ƒ''&nbsp;∈&nbsp;''F''. This definition usually appears in the context of [[topological vector space]]s.

When ''X'' is compact, a set is uniformly equicontinuous if and only if it is equicontinuous at every point, for essentially the same reason as that uniform continuity and continuity coincide on compact spaces. 
Used on its own, the term "equicontinuity" may refer to either the pointwise or uniform notion, depending on the context.  On a compact space, these notions coincide.

Some basic properties follow immediately from the definition. Every finite set of continuous functions is equicontinuous.  The closure of an equicontinuous set is again equicontinuous. 
Every member of a uniformly equicontinuous set of functions is [[uniformly continuous]], and every finite set of uniformly continuous functions is uniformly equicontinuous.

=== Examples ===

*A set of functions with a common [[Lipschitz constant]] is (uniformly) equicontinuous. In particular, this is the case if the set consists of functions with derivatives bounded by the same constant.
*[[Uniform boundedness principle]] gives a sufficient condition for a set of continuous linear operators to be equicontinuous.
*A family of iterates of an [[analytic function]]  is  equicontinuous on the  [[Fatou set]].<ref>Alan F. Beardon, S. Axler, F.W. Gehring, K.A. Ribet : Iteration of Rational Functions: Complex Analytic Dynamical Systems. Springer, 2000; {{ISBN|0-387-95151-2}}, {{ISBN|978-0-387-95151-5}}; page 49</ref><ref>Joseph H. Silverman : The arithmetic of dynamical systems. Springer, 2007. {{ISBN|0-387-69903-1}}, {{ISBN|978-0-387-69903-5}}; page 22</ref>

=== Counterexamples ===

*The set of all Lipschitz-continous functions is not equicontinous, as the maximal Lipschitz-constant is unbounded.
*The sequence of functions f<sub>n</sub>(x) = arctan(nx), is not equicontinuous because the definition is violated at x<sub>0</sub>=0.