An extension of a theorem of Marcinkiewicz and Zygmund on differentiability
Let f be a measurable function such that at each point x of a set E, where k is a positive integer, λ > 0 and 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 exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative 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...