Extending Peano differentiable functions (Q2785333)

Let \(H=H_p\cup H_c\subset{\mathcal R}\) be a closed set with a perfect part \(H_p\) and countable part \(H_c\), \(H_p\cap H_c=\emptyset\). Suppose that \(H\) is not symmetrically porous at each point \(x\in H_p\) and that a function \(f:H\to {\mathcal R}\) is \(k\)-times Peano differentiable at each point \(x\in H\) \((k>1)\). Then there is an extension \(F:{\mathcal R}\to{\mathcal R}\) of \(f\) which is \(k\)-times Peano differentiable on \(\mathcal R\) whose Peano derivatives agree with those of \(f\) on \(H\).