On the complexity of continuous functions differentiable on cocountable sets (Q1047489)

The author defines a map \(T\mapsto F_t\) from the set of subtrees of~\(\mathbb{N}^{<\omega}\) to~\(C\bigl([0,1]\bigr)\) with the property that \(F_T\)~is differentiable if \(T\)~is well-founded and \(F_T\)~is non-differentiable at a perfect set of points if \(T\)~is not well-founded. This shows that whenever \(\mathcal{R}\)~is a family of countable sets that contains~\(\emptyset\) the set of functions whose set of points of non-differentiability belongs to~\(\mathcal{R}\) is \(\boldsymbol\Pi_1^1\)-hard and hence \(\boldsymbol\Pi_1^1\)-complete if co-analytic. This shows \(\boldsymbol\Pi_1^1\)-completeness of various sets in one fell swoop: the sets of continuous functions whose non-differentiability sets are empty, finite, countable, or countable \(G_\delta\)-sets respectively.