W. Rudin gives the following example $f(x)$.
Let $\varphi(x) = |x|$ on $[-1,1]$ and extend periodically to the whole real line with period $2$.
Now define
$ f(x) = \sum_{n=0}^\infty \left(\frac{3}{4}\right)^n\varphi(4^nx).$
How does the graph of this function, or its successive approximations, look like? Just to get an idea of how continuity happens, but not differentiability.