4
$\begingroup$

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??

  • 1
    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....2011-12-04
  • 0
    Seems circular to me.2011-12-04
  • 0
    @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?2011-12-04
  • 0
    Dear Travis, Your idea is correct. It is also discussed in this answer: http://math.stackexchange.com/a/87936/221 Regards,2011-12-04
  • 0
    @Srivatsan Upon further inspection, perhaps this may not be circular. The usual proof of Darboux's theorem relies heavily on the IVT, but [this](http://en.wikipedia.org/wiki/Darboux's_theorem_(analysis)) one seems to avoid it.2011-12-04
  • 5
    @Ragib: Dear Ragib, Darboux's theorem doesn't depend on IVT, since $f'$ need not be continuous. Regards,2011-12-04
  • 0
    @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

1 Answers 1