If is an open set in with a smooth boundary . If is a continous -function so that for some and all (i.e. is constant on the boundary). And if for all . Then must be a constant function.
This is true by the maximum principle of harmonic functions. The maximum principle states that attains its maximum and minimum on the boundary. But since the function on the boundary is konstant it means for all . And so is constant.
Though, I am not sure if the maximum principle has any limitations. If happens to be bounded then it applies. What happens if is the right-half plane? I do not know in that case. If the maximum principle still applies then the argument still works.