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