The question's like this：
Show that a nonempty set E of real numbers is an interval if and only if every continuous real-valued function on E has an interval as its image.
Thank you so much for helping me out!
The statement is an if and only if. So, first prove one direction, then the other. To prove the first direction, assume that E is an interval. Then show that it follows that every continuous real-valued function on E has an interval as its image.
Do you need help with that proof? To get you started: If the function is a constant function, then it is trivially true, as every constant $\displaystyle c$ can be expressed as the interval $\displaystyle \[c,c\]$.
Next, assume that every continuous real-valued function on E has an interval as its image. Try to prove that E is an interval. This should be trivial to prove. Hint: Let $\displaystyle f: E \to E$ be the identity function.