Can I generate a continuous and non-differentiable function with basic calculus tools? Is there a simple way of expressing such a function?
The function $|x|$ is (uniformly) continuous everywhere, but not differentiable at $0$, because it has a corner. Finite sums of this function can give continuous maps which are not differentiable at any given finite set, and careful scaling can extend this to countable sets.
A continuous function which is nowhere differentiable is harder to construct, but quite doable with infinite series.