Editing Uniform continuity (section)

== Local continuity versus global uniform continuity ==
In the definitions, the difference between uniform continuity and [[Continuous function|continuity]] is that, in uniform continuity there is a globally applicable <math>\delta</math> (the size of a neighbourhood in <math> X </math> over which values of the metric for function values in <math> Y </math> are less than <math>\varepsilon</math>) that depends on only <math>\varepsilon</math> while in continuity there is a locally applicable <math>\delta</math> that depends on the both <math>\varepsilon</math> and <math>x</math>. Continuity is a ''local'' property of a function — that is, a function <math>f</math> is continuous, or not, at a particular point <math>x</math> of the function domain <math>X</math>, and this can be determined by looking at only the values of the function in an arbitrarily small neighbourhood of that point. When we speak of a function being continuous on an [[interval (mathematics)|interval]], we mean that the function is continuous at every point of the interval. In contrast, uniform continuity is a ''global'' property of <math>f</math>, in the sense that the standard definition of uniform continuity refers to every point of <math>X</math>. On the other hand, it is possible to give a definition that is ''local'' in terms of the natural extension <math>f^*</math>(the characteristics of which at nonstandard points are determined by the global properties of <math>f</math>), although it is not possible to give a local definition of uniform continuity for an arbitrary hyperreal-valued function, see [[Uniform continuity#Non-standard analysis|below]].

A mathematical definition that a function <math>f</math> is continuous on an interval <math>I</math> and a definition that <math>f</math> is uniformly continuous on <math>I</math> are structurally similar as shown in the following.

Continuity of a function <math>f:X \to Y</math> for [[metric spaces]] <math> (X,d_1) </math> and <math> (Y,d_2) </math> at every point ''<math>x</math>'' of an interval <math>I \subseteq X</math> (i.e., continuity of <math>f</math> on the interval <math>I</math>) is expressed by a formula starting with [[Quantification (logic)|quantifications]]
: <math>\forall x \in I \; \forall \varepsilon > 0 \; \exists \delta > 0 \; \forall y \in I :  \, d_1(x,y) < \delta \, \Rightarrow \, d_2(f(x),f(y)) < \varepsilon</math>,
(metrics <math>d_1(x,y) </math> and <math>d_2(f(x),f(y)) </math> are <math>|x - y| </math> and <math>|f(x) - f(y)| </math> for <math>f:\mathbb{R} \to \mathbb{R} </math> for [[Real number|the set of real numbers]] <math>\mathbb{R} </math>).

For uniform continuity, the order of the first, second, and third [[Quantifier (logic)|quantifications]] (<math>\forall x \in I </math>, <math>\forall \varepsilon > 0</math>, and <math>\exists \delta > 0</math>) are rotated:
: <math>\forall \varepsilon > 0 \; \exists \delta > 0 \; \forall x \in I \; \forall y \in I : \, d_1(x,y) < \delta \, \Rightarrow \,d_2(f(x),f(y)) < \varepsilon </math>.
Thus for continuity on the interval, one takes an arbitrary point <math>x</math> of the interval'','' and then there must exist a distance <math>\delta</math>,
: <math>\cdots \forall x \, \exists \delta \cdots ,</math>
while for uniform continuity, a single <math>\delta</math> must work uniformly for all points <math>x</math> of the interval,
: <math>\cdots \exists \delta \, \forall x \cdots .</math>