How to prove that a function is continuous in functional equation?

Question

I was wondering about different methods or properties to prove that a function in a functional equation is continuous or differentiable. Can somebody give me some examples of such problems or methods, where in the given functional equation you can prove that the function(s) are continuous and a proof of that? I know that if we find the function, it might happen to be continuous, but I'm looking for problems which first prove continuity before finding the solution.