I don't think your professor expects you to do all that, but if he does, you need to crank out the - proofs for each one.
For let , because
For , you can probably Google a proof. The idea is to let and then use a difference-to-product identity to convert into something more manageable.
is kind of a pain in the ass, and I've screwed up continuity proofs of this function before. There are probably other threads on MHF with proofs of this.