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I'm working on a project in Real Analysis and I am stuck on the following question:

Outline a proof, beginning with basic properties of the real numbers, of the following theorem - if f : [a,b] → (−∞,∞) is a continuous function such that f′(x) = 0 for all x ∈ (a,b), then f(a) = f(b).

I recognize this as an application of the Mean Value Theorem and have tried working backwards from that to get to some basic properties of real numbers but that method seems to be extremely slow. Does anyone have tips on how to answer this kind of question?

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  • $\begingroup$ it is not hard to prove a little more using MVT: math.stackexchange.com/questions/1516649 $\endgroup$
    – 311411
    Commented Apr 13, 2021 at 18:05
  • $\begingroup$ Yeah I can prove it, but how do I actually write a proof outline starting with basic properties of real numbers? $\endgroup$
    – Michael Burton
    Commented Apr 13, 2021 at 18:07
  • $\begingroup$ You want a very detailed proof, it seems. I was not sure what "proof outline" means; that to me sounds like a quick proof without much detail. Maybe check with instructor about exactly what is required? $\endgroup$
    – 311411
    Commented Apr 13, 2021 at 18:09
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    $\begingroup$ @MichaelBurton are you not allowed to just use the MVT? $\endgroup$
    – Chubby Chef
    Commented Apr 13, 2021 at 18:14
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    $\begingroup$ Does this mean you have to develop the notions of limit, continuity, derivative? $\endgroup$
    – saulspatz
    Commented Apr 13, 2021 at 18:56

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