SCHWEITZER 1982 Problem 5

Find a perfect set $H \subset [0,1]$ of positive measure and a continuous function $f$ defined on $[0,1]$ such that for any twice differentiable function $g$ defined on $[0,1]$ , the set $\{ x \in H : f(x)=g(x)\ \}$ is finite.

M. Laczkovich