Construct a continuous function of bounded variation on $[0,1]$ which is not monotone in any subinterval.
We can follow the pattern of the Cantor-Lebesgue function (somewhat). For example, at the first stage, let the function increase to $1/2 + \epsilon$, then decrease by $2\epsilon$, and then increase again by $1/2 + \epsilon$. I want to choose the $\epsilon$'s at each stage so that their sum converges but I'm not sure how to continue with this idea.
There is a rather nice literature on "nowhere monotonic" functions with a number of interesting questions.
You can even construct an everywhere differentiable function with a bounded derivative that is nowhere monotone.
Search under that topic. You should find at least the discussion here:
https://mathoverflow.net/questions/167323/everywhere-differentiable-function-that-is-nowhere-monotonic