On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

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We comment on a Mazur problem from "Scottish Book" concerning second partial derivatives. It is proved 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 Fy(x):=f'y(x,y) has finite variation, then almost everywhere on the rectangle there exists the partial derivative f"yx. We construct a separately twice differentiable function, whose partial derivative f'ₓ is discontinuous with respect to the second variable on a set of positive measure. This solves in the negative the Mazur problem.