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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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    @Matt indeed my memory has failed me yet again. I just looked at Hardy's "A course in pure mathematics" where I thought I saw the proof invoking IVT, but indeed it only uses the extreme value theorem.2011-12-04

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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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    I was proving the sometimes called the Mean Value Theorem for integrals.2011-12-04