Continuity And Range

Question

Can somebody please check my solution for this question?

Answer 1

Your two first answers are correct. The third one is not. Consider, for instance,$$f(x)=\frac12+\frac12(1-x)\sin\left(\frac\pi x\right).$$Its range is $(0,1)$.

Answer 2

Your first two answers are correct. Reasoning is kind of unnecessary.

(a) Extreme value theorem posits on a compact interval, all extreme values are achieved by the function.

(b) Your example shows this can happen.

(c) (0,1] can contain a compact set, say $[\epsilon, 1]$, which becomes the case of the first problem. But that doesn’t stop you from considering an open set $(0,1-\epsilon)$ and then tacking on a continuous extension at the end.