The main result of this paper is that if f is n-convex on a measurable subset E of R, then f is n - 2 times differentiable, n - 2 times Peano differentiable and the corresponding derivatives are equal, and f((n-1)) = f((n-1)) except on a countable set. Moreover f((n-1)) is approximately differentiable with approximate derivative equal to the nth approximate Peano derivative of f almost everywhere.

Let A be a subset of a Banach space X and f a Frechet differentiable function on A (with respect to A). We give a simple proof of the connectedness of the graph of f' in X x X* under relatively weak conditions on A. In particular, we simplify a proof by J. Maly of the connectedness of the range of f' for some convex sets A. At the same time, we extend an older result of C. E. Weil on the connectedness of the range of f' for some non-convex sets A subset of R-n.