If a real function f is differentiable on a perfect subset H of the real line, then f' is Baire 1 on H and f can be extended to R as an everywhere differentiable function. The authors have studied similar questions for functions of several variables. Among others they have proved that in the case of arbitrary closed set H whenever the derivative is determined uniquely, it must be Baire 2. If the tangent space of H is sufficiently rich, then the derivative is Baire 1. A function defined on H can be extended to a function which is everywhere differentiable on \(R^ n\) if and only if its derivative is Baire 1.