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.
The Weierstrass function W defined as by
where 0 < H < 1 and ω > 1 is continuous and nondifferentiable
for all values of x. This function is well defined, for example, to 100
digits, we may calculate
However, Figure 1 shows a zoom on the Weierstrass function W0.3, 1.3(x)
starting from the interval [0, 1] and slowly zooming by over six orders of magnitude.
The red point is (0.5, W0.3, 1.3(0.5)).
Figure 1. A zoom on the point (0.5, W0.3, 1.3(0.5)).
Compare the zoom on the Weierstrass function with the zoom demonstrated
on the differentiable functions page.