Differentiable extension theorem: A lost proof of V. Jarník

We present a beautiful but relatively unknown theorem that every differentiable function f:P→R, with P⊂R being closed, admits differentiable extension F:R→R. We present an elementary proof of this result based on a construction sketched in a hard-to-access 1923 paper [4] of V. Jarník. Using this construction, we also obtain an elegant version of Whitney extension theorem characterizing when such an f admits continuously differentiable extension.