I do not know how to solve below problem; maybe someone can guide me through it:
Prove that is continuous on for any positive integer .
It's easy to show that is not only continuous, but uniformly continuous on every closed segment . Indeed, fix a and assume . We have
(this is checked by multiplying the right-hand side). The second factor in the right-hand side has terms and . Therefore, . The expression is a constant for each fixed , so a small variation in produces only a small variation in . It's easy to turn this into an proof.