# Intuitive proof of the Intermediate value theorem to first year calculus students

I want to give an informal proof of the intermediate value theorem to my calculus students. They have just learned about continuity and they don't know yet about derivatives among more other advanced topics.

Do you think it's possible?

• Just draw a horizontal line and a point below and a point above and clearly to connect the points you have to cross the line. Since $\Bbb R$ has no holes it has to hit it. Dec 7 '17 at 3:08
• Roughly speaking a function is continuous if there are no jumps in its graph, so its like drawing it continuously without lifting the pencil so you will eventually cover all points in between.
– user428700
Dec 7 '17 at 3:09
• I'd be wary of it doing more harm than good. I'm not sure there's anything more mysterious than a proof of something "obviously true" (assuming the IVT is sufficiently obviously true to students, which, I think, it generally is). There will be more opportunities to sketch proofs of things that are not quite as obvious, like the MVT, l'Hospital's rule, or the Fundamental Theorem of Calculus. Dec 7 '17 at 3:20
• Unless your students are aware of the properties of real numbers I don't think a proof for IVT can be explained to them. Dec 7 '17 at 3:26
• Unfortunately the high school curriculum for mathematics in most countries does not include the pre-requisites for understanding calculus and therefore teaching calculus becomes a very difficult task. You may perhaps include a small half an hour lecture on two essential topics : 1) density of rationals (and real numbers) 2) an introduction to completeness of real numbers (proofs may be omitted here). IVT is a cakewalk after this. Dec 7 '17 at 6:02

## 2 Answers

Assume that $f(a)<c<f(b)$, bisect the interval $[a,b]$ into $[a_{2},b_{2}]$ such that $f(a_{2})<c<f(b_{2})$ (if this fails, that means we have reached the goal). Proceed this again to get $[a_{3},b_{3}]$... so the left endpoints and right endpoints converge to a point $\eta$, continuity then implies $f(\eta)=c$.

First year was four years ago for me, and I remember being rather overwhelmed by analysis the first time I met it. Having said that, I wasn't a particularly remarkable student, and it also depends on where it is that you're teaching.

My point is, it depends on what you want your informal proof to include. I'm pretty convinced you could provide a formal proof and they'd be able to understand - it may just take a bit of hard work. Perhaps they don't have the tools they need for it?

An informal proof could be anything from drawing a line between two points, to providing a reasonable sketch of the proper proof. I think an intuitive understanding of the intermediate value theorem isn't difficult for first years, so perhaps you can strive for a reasonable sketch proof.