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Let f(x) be monotone increasing. Define $S=\{x|c \leq f(x) \leq d\}$. Show that S must be a single interval.

I understand this intuitively and graphically, but don't know how to prove this formally. Please help with a proof strategy.

kiwifruit
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Hint: you need to show that if $x_1\in S$ and $x_3\in S$ and $x_1<x_2<x_3$, then $x_2\in S$.

David
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  • Hmm, ok, but would this prove that there is only one interval, and not multiple ones such as for sin or cos? – kiwifruit Feb 04 '14 at 04:18
  • Yes! Draw two pictures on the real line to see why: (1) an interval; (2) two separate intervals. Is the statement I suggested you prove true in both cases? – David Feb 04 '14 at 04:26
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    Hmm, but if I take x1 and x2 from the same interval and a larger x3 from another repeating part of the interval, wouldn't the result also hold? – kiwifruit Feb 04 '14 at 04:35
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    How could that happen? Give me an example. I don't think I understand what you mean. – David Feb 04 '14 at 04:40
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    Something like this: enter image description here – kiwifruit Feb 04 '14 at 04:54
  • But that doesn't always work: you could have taken $x_2$ in between the two blue parts. This proves that the blue parts do not form an interval. – David Feb 04 '14 at 05:05
  • Ok, I see, thank you! – kiwifruit Feb 04 '14 at 14:52