The Baire Classification of Ordinary and Approximate Partial Derivatives

Introduction. It is well known that a function which is a derivative is in the first Baire class, and it is also known that a function which is an approximate derivative is in the first Baire class. The analogous statement for partial derivatives of functions of more than one variable is of course not true. For if g is a nonmeasurable function of one variable, then the function f(x, y)=x-g(y) has a partial derivative which is nonmeasurable. Thus it is seen that a function of n variables need be restricted somewhat in order for its partial derivative to be in the first Baire class. G. P. Tolstov [4] has shown that if /is linearly continuous on a domain GCZR2 and if df/dx exists everywhere in G, then df/dx is in the first Baire class. In addition, he has given an example of a linearly continuous function of three variables whose partial derivative exists and is not in the first Baire class. In this paper we show that Tolstov's theorem remains true if approximate partial differentiation is used in place of ordinary partial differentiation.