Let . Prove is continuous on .
NOTE: for this proof, you're allowed to assume is continuous on .
It is a very interesting question. It looks very similar to the Weierstrass function (which is differenciable nowhere but continous).
Note: In complex analysis a beautiful consquence happens, namely say are analytic on converging uniformly to then is analytic on (Weierstrass theorem, I think that is how it is called).