THEOREM (SIERPINSKI). If 2No = P;l, there exists an infinite sequence
Frederick Bagemihl, H. D. Sprinkle · 2016
of single-valued functions of a real variable fo(x), fi(x), f2(x), * * * such that, for every nondenumerable set N of real numbers, all the functions of the sequence, except perhaps a finite number of them, transform N onto the set of all real numbers. An examination of the direct proof shows, however, that all that can be said about the sequence of functions fo(x), fi(x), f2(x), * * * defined there is that infinitely many of them, but not necessarily all except perhaps a finite number, transform N onto the set of all real