The Baire classification of partial derivatives of functions of two variables
by
Volodymyr Mykhaylyuk
Chernivtsi National University

In the talk we shall discuss the Baire classification of partial derivatives of separately differentiable functions.

G. Tolstov [1] proved that the partial derivative f'_x of a separately continuous function f:R×R→R, which is differentiable with respect to the first variable, is a function of the first Baire class. L. Snyder [2] generalized this result to approximative partial derivatives of functions of two real variables.

Theorem 1 Let n ∈ N, X=R, Y be a topological space, f:X×Y→R be a function which is continuous with respect to the second variable and such that E={(x, y) ∈ X×Y: ∃  [(∂ⁿ f)/(∂xⁿ)](x, y)}. Then the function g:E→R, g(x, y)=[(∂ⁿ f)/(∂xⁿ)](x, y) is of the first Baire class.

References.

[1] Tolstov G. On partial derivatives // Izv. Akad. Nauk SSSR Mat. 13 (1949), 425-449.

[2] Snyder L. The Baire classification of ordinary and approximate partial derivatives // Proc. Amer. Math. Soc. 17:1 (1966), 115-123.

Date received: May 6, 2008