The continuity of functions of many variables
Richard B. Kershner · Transactions of the American Mathematical Society · 1943
Introduction.It is known that a function f(x, y) of two real variables may be continuous with respect to each variable separately throughout a given region without being continuous with respect to (x, y) at all points of the region.In fact, W. H. and G. C. Young(x) have given an example of a function f(x, y) which is a continuous function of the position along every straight line in the unit square [0, l]X [0, l] but which has an uncountable number of two-dimensional discontinuities in every rectangle contained in the unit square.The example of W. H. and G. C. Young could easily be modified so as to yield a function continuous along every analytic arc but with an uncountable number of discontinuities in every rectangle.If the number of variables is greater than two the situation becomes even worse.As was pointed out by Baire( 2), for three variables, and subsequently by Hahn(3), for any number of variables, a function/(xi, x2, • • • , xn) may be continuous in each variable xt-and yet be discontinuous with respect to (xi, X2, • • • , xn) at every point of an (« -2)-dimensional rectangle.In fact let g{xi, Xi) be a function continuous in xi and x2 separately but discontinuous at (0, 0).Then /{*!, Xn) = g(Xl, X2) is discontinuous at every point of the (» -2)-dimensional region xi = 0, Xa = 0. Finally, Lebesgue(4) has shown that a function/(xi, x2, • • • , xn) which is continuous in each variable x< separately may be of the (n-l)st Baire class, although no worse.The problem of considering how much could be said concerning the w-dimensional continuity points of a function /(xi, x2, • • • , x") which is assumed to be continuous in each x,-separately was first treated in 1899 by Baire in the fundamental paper(6) in which he introduced most of the classic notions associated with his name.For the case of two variables his results were complete.He showed that