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.