Abstract Let 𝐸 be a set consisting of rectangular frames in . The following question connected with one problem of A. Zygmund is studied in the paper: Does there exists a function the integral of which is: 1) non-differentiable almost everywhere in a strong sense along every frame from 𝐸, and 2) strongly differentiable along every frame not belonging to 𝐸? In particular, the question is solved on the existence of a non-empty set 𝐸 different from the set of all rectangular frames for which there is a function with the properties 1) and 2).

This paper demonstrates that a function can be strongly differentiable along every frame in a set E, while being non-differentiable almost everywhere along every frame in E.

On the Strong Differentiation of Multiple Integrals along Different Frames

Key Takeaway: This paper demonstrates that a function can be strongly differentiable along every frame in a set E, while being non-differentiable almost everywhere along every frame in E.