Editing Uniform continuity (section)

=== Definition of uniform continuity ===

* <math>f</math> is called '''uniformly continuous''' if for every [[real number]] <math> \varepsilon > 0 </math> there exists a real number <math> \delta > 0 </math> such that for every <math> x,y \in X </math> with <math> d_1(x,y) < \delta </math>, we have <math> d_2(f(x),f(y)) < \varepsilon </math>. The set <math> \{ y \in X: d_1(x,y) < \delta\} </math> for each <math> x </math> is a neighbourhood of <math> x </math> and the set <math> \{ x \in X: d_1(x,y) < \delta\} </math> for each <math> y </math> is a neighbourhood of <math> y </math> by [[Neighbourhood (mathematics)|the definition of a neighbourhood in a metric space]].
** If <math> X </math> and <math> Y </math> are subsets of the [[real line]], then <math> d_1 </math> and <math> d_2 </math> can be the [[Real_line#As_a_metric_space|standard one-dimensional Euclidean distance]], yielding the following definition: for every real number <math> \varepsilon > 0 </math> there exists a real number <math> \delta > 0 </math> such that for every <math> x,y \in X </math>, <math> |x - y| < \delta \implies |f(x) - f(y)| < \varepsilon </math> (where <math> A \implies B </math> is a [[material conditional]] statement saying "if <math> A </math>, then <math> B </math>").
* Equivalently, <math>f</math> is said to be uniformly continuous if <math>\forall \varepsilon > 0 \; \exists \delta > 0 \; \forall x \in X \; \forall y \in X : \, d_1(x,y) < \delta \, \Rightarrow \,d_2(f(x),f(y)) < \varepsilon </math>. Here [[Quantification (logic)|quantifications]] (<math>\forall \varepsilon > 0  </math>, <math>\exists \delta > 0  </math>, <math>\forall x \in X  </math>, and <math>\forall y \in X  </math>) are used.
* Equivalently, <math>f</math> is uniformly continuous if it admits a [[modulus of continuity]].