# Interval and its image

• Oct 11th 2011, 05:37 PM
violet8804
Interval and its image
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!
• Oct 12th 2011, 01:50 AM
SlipEternal
Re: Interval and its image
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 $c$ can be expressed as the interval $$c,c$$.

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 $f: E \to E$ be the identity function.