Let be continuous with . Prove that if is not of the form , then there does not necessarily exist satisfying . Provide an example that illustrates this using .
So I was given the hint that I can use a modified function, however I'm not really sure how I would go about that. Preferably, an example not using that would be great.
My thoughts so far are that I use a proof by contradiction saying that such that and . And I need to get to the point where , which would create the contradiction (I assume that's the endpoint?). However, how would I use to prove that? Is it trivial in that is false? Also, it isn't given that , so maybe I shouldn't assume that?