  Given a compact Riemann surface X and a point x_0 in X, we construct a
holomorphic function without critical points on the punctured Riemann surface R
= X - x_0 which is of finite order at the point x_0. This complements the
result of Gunning and Narasimhan from 1967 who constructed a noncritical
holomorphic function on every open Riemann surface, but without imposing any
growth condition. On the other hand, if the genus of X is at least one, then we
show that every algebraic function on R admits a critical point. Our proof also
shows that every cohomology class in H^1(X;C) is represented as a de Rham class
by a nowhere vanishing holomorphic one-form of finite order on the punctured
surface X-x_0.
