Hi all,

I need to prove the following lemma in attachment . Our instructor wants us to do it in the following way:

Let A be the set of all with the property that there exists a continuous function

such that

and

Prove that A is nonempty , closed and relatively open in [0,1] and conclude that A = [0,1] since [0,1] is connected.

Can anyone help?? I don't even have a slight idea how to proceed.