Here is an example of a discontinuous function. It has a [i]jump discontinuity[/i] at [math]x=1[/math]. You can move [math]a[/math] and alter [math]\delta[/math], the half-width of the interval in [math]x[/math], in order to see how [math]\varepsilon[/math], the half-width of the interval in [math]f\left(x\right)[/math], changes. Try moving [math]a[/math] on top of a discontinuity, and see how the behaviour of the box changes.

This is a different case of a discontinuous function: we have defined [math]f\left(1\right)=0[/math], but there is no limit as [math]x\longrightarrow1[/math]. This is called an [i]essential discontinuity[/i].

Other types of discontinuity exist, but these two illustrate the most common.[br][br]Something commonly thought of as a discontinuity, though in fact not, is the point [math]x=0[/math] in the function [math]f\left(x\right)=\frac{1}{x}[/math]. In fact, [math]f[/math] is not defined at this point, and so the point is called a singularity. If we were to define a new function, equal to [math]\frac{1}{x}[/math] when [math]x\ne0[/math] and equal to [math]0[/math] when [math]x=0[/math], then such a function would be defined at [math]x=0[/math] (obviously), but the function would still not be continuous at this point, as [math]\lim_{x\longrightarrow0}\frac{1}{x}[/math] does not exist. It would be another example of an essential discontinuity.