An extension of a theorem of Marcinkiewicz and Zygmund on differentiability

Let f be a measurable function such that ${\displaystyle {\Delta }_k(x,h;f)=O\left({\left|h\right|}^{\lambda }\right)}$ at each point x of a set E, where k is a positive integer, λ > 0 and ${\displaystyle {\Delta }_k(x,h;f)}$ 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 ${\displaystyle f_{\left(k\right)}}$ exists a.e. on E. Here we prove that if λ > k-1 then the Peano derivative ${\displaystyle f_{\left([\lambda ]\right)}}$ 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 ${\displaystyle f_{(\lambda ),a}}$ exists on E then ${\displaystyle f_{(\lambda )}}$ exists a.e. on E.