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Let f be a measurable function such that ∆k(x, h; f) = O(|h|λ) at each point x of a set E, where k is a positive integer, λ > 0 and ∆k(x, h; f) is the symmetric difference of f at x of order k. Marcinkiewicz and Zygmund [5] proved that if λ = k and if E is measurable then the Peano derivative f(k) exists a.e. on E. Here we prove that if λ > k − 1 then the Peano derivative f([λ]) exists a.e. on E and that the result is false if λ = k − 1; it is further proved that if λ is any positive integer and if the approximate Peano derivative f(λ),a exists on E then f(λ) exists a.e. on E.

An extension of a theorem of Marcinkiewicz and Zygmund on differentiability