I said that this statement was false and used the counterexample

and

I put true.

Since g is continuous I know that

.

f is also continuous so

.

Let

Define

. WLOG, let

.

**is this step valid? **
This gives

as required.

**Since my initial definitions only apply to a rational c, would I need to prove they are equal when c is not rational? **
I opted for true.

f is continuous:

.

g bounded:

where

.

Let

Since I am only interested in the behaviour of the multiple functions at 0, pick

.

**can you do this? I need to restrict delta somehow so large values of delta can also be used.**
This gives that:

as required.

I said this is true since the composition of two continuous functions is itself continuous. However, I cannot formulate a proof!

I have a feeling this is false. Unfortunately I am unable to think of a counterexample.

Any help with any of these questions would be appreciated. They get so much harder towards the end!