Entire functions mapping countable dense subsets of the reals onto each other monotonically

Abstract

It is shown that for arbitrary countable dense ssets A and B of the real line, there exists a transcendental entire function whose restriction to the real line is a real-valued strictly monotone increasing surjection taking A onto B The technique used is a modification of the procedure Maurer used to show that for countable dense subsets A and B of the plane, there exists a transcendental entire function whose restriction to A is a bijection from A to B .