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

We show that the Riemann zeta function \zeta\ has only countably many self-intersections on the critical line, i.e., for all but countably many z in C the equation \zeta(1/2+it)=z has at most one solution t in 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) in R^2:a \ne b and F(a)=F(b)} is countable, or the image F(R) is a loop in C.