I was thinking about the following problem:
Let $f:\mathbb R \rightarrow \mathbb R$ be continuous with $f(0)=f(1)=0.$Then which of the following is not possible?
(a) $f([0,1])=\{0\},$
(b) $f([0,1])=[0,1),$
(c) $f([0,1])=[0,1],$
(d) $f([0,1])=[-1/2,1/2].$
Can someone point me in the right direction? Thanks in advance for your time.