# Differentiability of Regulated Functions

In general, what can be said about differentiability of (real-valued) regulated functions, i.e. such for which the left and right limit exist at every point?

Such functions are necessarily continuous except at countably many points, but "how many" points of differentiability do they have? Are they differentiable almost everywhere? Is there a nowhere differentiable regulated function?

A quick google search doesn't seem to help..

$$\sum_{k=1}^{\infty} a^k \cos (b^k \pi x)$$
Where we take $0<a<1$ (in order to make the series converge) and $ab$ sufficiently large; for further references, look for Weierstrass's function on the web. He was the first to discover this kind of phenomena.