An example of a convex function $f:[0,1]\to \mathbb{R}$ which is not differentiable at infinitely many $x\in[0,1]$.

Question

An example of a convex function $f:[0,1]\to \mathbb{R}$ which is not differentiable at infinitely many $x\in[0,1]$.
I was thinking about Dirichlet function but it is not convex, according to this: https://math.stackexchange.com/a/1816062/270833

Answer 1

Take a particular convex function that is not differentiable at $x=0$, say $$ f(x) = \begin{cases} 0, &\text{if } x\le 0, \\ x, &\text{if } x\ge 0, \end{cases} $$ and add together infinitely many relatives of that function, for example as follows: $$ g(x) = \sum_{n=2}^\infty \frac1{2^n} f\bigg( x-\frac1n \bigg). $$ The coefficient $\frac1{2^n}$ ensures that the series converges uniformly on $[0,1]$ and thus is itself a continuous convex function; however, $g(n)$ is not differentiable at any of $\{\frac12,\frac13,\frac14,\dots\}$.