Note on H-convex functions.
Given an open subset of and a Borel function , conditions on are given which assure the lower semicontinuity of the functional with respect to different topologies.
We comment on a problem of Mazur from “The Scottish Book" concerning second partial derivatives. We prove that if a function f(x,y) of real variables defined on a rectangle has continuous derivative with respect to y and for almost all y the function has finite variation, then almost everywhere on the rectangle the partial derivative exists. We construct a separately twice differentiable function whose partial derivative is discontinuous with respect to the second variable on a set of positive...