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
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 (1/2½, χ(1/2½))
= (1/2½, 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 1/2½.