Editing Uniform continuity (section)

=== Definition of (ordinary) continuity ===

* <math>f</math> is called '''continuous''' <math> \underline{\text{at } x} </math> if for every [[real number]] <math> \varepsilon > 0 </math> there exists a real number <math> \delta > 0 </math> such that for every <math> 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> is a neighbourhood of <math> x </math>. Thus, (ordinary) continuity is a local property of the function at the point <math> x </math>.
* Equivalently, a function <math>f</math> is said to be continuous if <math>\forall x \in X \; \forall \varepsilon > 0 \; \exists \delta > 0 \; \forall y \in X :  \, d_1(x,y) < \delta \, \Rightarrow \, d_2(f(x),f(y)) < \varepsilon</math>.
* Alternatively, a function <math>f</math> is said to be continuous if there is a function of all positive real numbers <math>\varepsilon</math> and <math>x \in X</math>, <math>\delta(\varepsilon, x)</math> representing the maximum positive real number, such that at each <math>x</math> if <math>y \in X</math> satisfies <math> d_1(x,y) < \delta(\varepsilon,x) </math> then <math> d_2(f(x),f(y)) < \varepsilon </math>. At every <math>x</math>, <math>\delta(\varepsilon, x)</math> is a monotonically non-decreasing function.