Self-intersections of the Riemann zeta function on the critical line

We show that the Riemann zeta function ζ has only countably many self-intersections on the critical line, i.e., for all but countably many z ∈ C the equation ζ ( 1 2 + i t ) = z has at most one solution t ∈ R . More generally, we prove that if F is analytic in a complex neighborhood of R and locally injective on R , then either the set { ( a , b ) ∈ R 2 : a ≠ b  and  F ( a ) = F ( b ) } is countable, or the image F ( R ) is a loop in C .