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I am studying for a final and have been solving extra problems in Spivak's Calculus. However, I am not sure how to write out the proof for a star problem in Chapter 14: Fundamental Theorem of Calculus.

It reads:

Use the Fundamental Theorem of Calculus and Darboux's Theorem to give a proof of the Intermediate Value Theorem.

I think I have the idea, but I can't seem to formulate it into a rigorous formal proof. Basically, what I have in mind is letting $F$ be a function such that $F'=f$, and applying Darboux's thoerem on $F$. Then by FTC, we have the IVT??

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Yep, you have the right idea. If $f$ is continuous, by FTC you can find $F$, and Darboux Theorem tells you that $F'=f$ verifies the IVT... I am not sure thought that this is not a circular proof.... –  N. S. Dec 4 '11 at 3:47
    
Seems circular to me. –  Ragib Zaman Dec 4 '11 at 4:01
    
@Ragib I have have my doubts about circularity, but I cannot point out precisely where. If you're able to see it, can you explain where? –  Srivatsan Dec 4 '11 at 4:07
    
Dear Travis, Your idea is correct. It is also discussed in this answer: math.stackexchange.com/a/87936/221 Regards, –  Matt E Dec 4 '11 at 4:15
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@Ragib: Dear Ragib, Darboux's theorem doesn't depend on IVT, since $f'$ need not be continuous. Regards, –  Matt E Dec 4 '11 at 4:18
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1 Answer

Let $a\lt b$, $f:[a,b]\to \mathbb{R}$, continuous. Let $[x_1, x_2]\subset (a,b)$, let $c$ between $f(x_1)$ and $f(x_2)$, wlog $f(x_1)\lt c\lt f(x_2)$. By the FTC, this says that $$F'(x_1)\lt c\lt F'(x_2),$$ where $F:[a,b]\to\mathbb{R}:x\mapsto \int_a^x f(t) dt$. By Darboux's Theorem $F'$ must satisfy the intermediate values property, then there exist $u\in (x_1,x_2)$ such that $F'(u)=c$, i.e. $f(u)=c$.

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Typo: "Barboux" should read "Darboux". –  Matt E Dec 4 '11 at 4:18
    
Thanks for catch it @MattE –  leo Dec 4 '11 at 4:26
    
I was proving the sometimes called the Mean Value Theorem for integrals. –  leo Dec 4 '11 at 4:56
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