A function can still be continuous and not be differentable at a point. For
example, the absolute value function |*x*| is continuous at *x* = 0, but
the deriviatve is not defined at that point. Here we give an example of a
function, the Weierstrass function W, which is continuous everywhere but differentiable nowhere.

A function need not be continuous anywhere, for example, the
function:

where **Q** is the set of rationals.

It can also get worse: Thomae's function χ(*x*) is a function which is
continuous at all irrationals and discontinuous at all rationals:

Figure 1 shows a plot of the function χ(t) while Figure 2 shows a
zoom on the black point (1/2^{½}, χ(1/2^{½}))
= (1/2^{½}, 0) (because the square root of two is irrational).
If the points in Figure 1 are too fine, you may also look at this view.

Figure 1. Thomae's function.

Figure 2. A zoom on Thomae's function at 1/2^{½}.