Editing Uniform continuity (section)

== Visualization ==
For a uniformly continuous function, for every positive real number <math>\varepsilon > 0</math> there is a positive real number <math>\delta > 0</math> such that two function values <math>f(x)</math> and <math>f(y)</math> have the maximum distance <math>\varepsilon</math> whenever <math>x</math> and <math>y</math> are within the maximum distance <math>\delta</math>. Thus at each point <math>(x,f(x))</math> of the graph, if we draw a rectangle with a height slightly less than <math>2\varepsilon</math> and width a slightly less than <math>2\delta</math> around that point, then the graph lies completely within the height of the rectangle, i.e., the graph do not pass through the top or the bottom side of the rectangle. For functions that are not uniformly continuous, this isn't possible; for these functions, the graph might lie inside the height of the rectangle at some point on the graph but there is a point on the graph where the graph lies above or below the rectangle. (the graph penetrates the top or bottom side of the rectangle.)

<gallery widths="400" heights="300">
File:Gleichmäßig stetige Funktion.svg|For uniformly continuous functions, for each positive real number <math>\varepsilon > 0</math> there is a positive real number <math>\delta > 0</math> such that when we draw a rectangle around each point of the graph with a width slightly less than <math>2\delta</math> and a height slightly less than <math>2\varepsilon</math>, the graph lies completely inside the height of the rectangle.
File:Nicht gleichmäßig stetige Funktion.svg|For functions that are not uniformly continuous, there is a positive real number <math>\varepsilon > 0</math> such that for every positive real number <math>\delta > 0</math> there is a point on the graph so that when we draw a rectangle with a height slightly less than <math>2\varepsilon</math> and a width slightly less than <math>2\delta</math> around that point, there is a function value directly above or below the rectangle. There might be a graph point where the graph is completely inside the height of the rectangle but this is not true for every point of the graph.
</gallery>