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Math Help - Interval and its image

  1. #1
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    Exclamation 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!
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  2. #2
    MHF Contributor
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    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.
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