Differentiable restrictions of continuous functions
Let \(f: P\to R\). If f is continuous, P is perfect and \(m(P)>0,\) then for every \(\epsilon >0\) there is a perfect \(Q\subset P\) such that \(m(P- Q)<\epsilon\) and \(f|_ Q\) is differentiable. If f is measurable, and \(m(P)>0,\) then either (1) there is a perfect \(Q\subset P\) such that \(Q\neq \emptyset\) and \(f|_ Q\) is infinitely differentiable on Q and \((f|_ Q)^{(n)}=0\) for n sufficiently large, or (2) for every \(\epsilon >0\) there is a perfect \(Q\subset P\) such that \(m(P-Q)<\epsilon\) and \(f|_ Q\) is infinitely differentiable on Q.
scientific article; zbMATH DE number 3889828