Now asked on MO here.
This question is a generalisation of a prior question. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of circles with radius $r$ required to cover the graph of $f$?
It is easy to prove (by using the extreme value theorem) that only finitely many circles are required to cover the graph of $f$.
But how can I find the least number of circles? I don't think a closed form exists (I also think Fourier series might be a part of the solution to this problem but I couldn't reach am algorithm using it), but is there another solution, like an indefinite integral? If there isn't, is there an algorithm that can solve this problem?