Editing Real analysis (section)

====Uniform continuity====
{{Main|Uniform continuity}}
'''Definition.''' If <math>X</math> is a subset of the [[real number]]s, we say a function <math>f:X\to\mathbb{R}</math> is '''''uniformly continuous''''' '''''on''''' <math>X</math> if, for any <math>\varepsilon > 0</math>, there exists a <math>\delta>0</math> such that for all <math>x,y\in X</math>, <math>|x-y|<\delta</math> implies that <math>|f(x)-f(y)| < \varepsilon</math>.

Explicitly, when a function is uniformly continuous on <math>X</math>, the choice of <math>\delta</math> needed to fulfill the definition must work for ''all of'' <math>X</math> for a given <math>\varepsilon</math>.  In contrast, when a function is continuous at every point <math>p\in X</math> (or said to be continuous on <math>X</math>), the choice of <math>\delta</math> may depend on both <math>\varepsilon</math> ''and'' <math>p</math>.  In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point <math>p</math> is meaningless.

On a compact set, it is easily shown that all continuous functions are uniformly continuous.  If <math>E</math> is a bounded noncompact subset of <math>\mathbb{R}</math>, then there exists <math>f:E\to\mathbb{R}</math> that is continuous but not uniformly continuous.  As a simple example, consider <math>f:(0,1)\to\mathbb{R}</math> defined by <math>f(x)=1/x</math>.  By choosing points close to 0, we can always make <math>|f(x)-f(y)| > \varepsilon</math> for any single choice of <math>\delta>0</math>, for a given <math>\varepsilon > 0</math>.