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

We show that the Riemann zeta function ζζ has only countably many self-intersections on the critical line, i.e., for all but countably many z∈Cz∈C the equation ζ(12+it)=z has at most one solution t∈Rt∈R. More generally, we prove that if FF is analytic in a complex neighborhood of RR and locally injective on RR, then either the set {(a,b)∈R2:a≠b  and  F(a)=F(b)}{(a,b)∈R2:a≠b  and  F(a)=F(b)} is countable, or the image F(R)F(R) is a loop in CC.