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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.