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??
Best 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. $f(u)=c$.