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It is shown that for arbitrary countable dense subsets 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 ^ is a bijection from A to B .doi:10.1017/s0004972700040636 fatcat:jqh2caftgret3k2cowwp5e5cvq