We construct examples of nonparametric surfaces z = h(x, y) of zero mean curvature which satisfy contact angle boundary conditions in a cylinder in R3 over a convex domain with corners. When the contact angles for two adjacent walls of the cylinder differ by more than −2 , where 2 is the opening angle between the walls, the (bounded) solution h is shown to be discontinuous at the corresponding corner. This is exactly the behavior predicted by the Concus–Finn conjecture. These examples currently constitute the largest collection of capillary surfaces for which the validity of the Concus–Finn conjecture is known and, in particular, provide examples for all contact angle data satisfying the condition above for opening angles 2 2 ( /2, ).