What to teach in a class about the intermediate value theorem I'm going to give a lecture in a public contest (like a civil service examination to be a university public teacher in my country) about the intermediate value theorem in 45 minutes. I'm going to give this lecture to three professors (the examiners) but it's like I'm going to give it to undergraduate students.
Since the proof of the intermediate value theorem is too hard to show to someone who's never learned real analysis, what should I do in this class? I'm open to suggestions.
 A: You should first start with some explanation of the theorem via graphic illustration and then present some key applications. Apart from those mentioned in the other answer one of the basic applications of intermediate value theorem is the existence of $n$'th roots of positive real numbers. Usually when roots are introduced in high school no one really discusses their existence.
Maybe you can also add some historical remarks about the theorem and the fact that the theorem was used much before it was finally proved. And in fact people did not even think it needed a proof. 
And then you can provide proofs based on multiple approaches. It is also important to stress that intermediate value property is not the privilege of continuous functions only and there are discontinuous functions which have this property and one important class of such functions is the class of derivatives.

Note: I am not an educator by profession and the above is not based on any teaching experience but rather based on what I would have liked in a class on this theorem if I were a student. 
A: You should mention some of its consequences, such as:


*

*every polynomial function from $\mathbb R$ into itself with odd degree has at least one root;

*if $a,b\in\mathbb R$ are such that $a^2+b^2=1$ then there is some $\theta\in[0,2\pi)$ such that $a=\cos\theta$ and $b=\sin\theta$.

A: Introduce the problem by stating a real world example, e.g.

This morning we had a temperatur of $10$ °C, and now its $15$ °C. So we conclude that there must have been a time between this morning and now where we had a temperature of $13$ °C.

But why can we conclude this? This is not a universal rule, e.g.

Yesterday my bank account contained $100\$$. Then I went to the bank and cashed out $50\$$, i.e. now I have $50\$$ left on my account. Was there a time between yesterday and now where my account had exactly $70\$$ on it?

No. Here you can conclude that the important property we need to derive such kind of intermediate statements is continuity.
Now you can go about to give a short definition of what continuity means (if necessary) and a general formulation of the intermediate value theorem (IVT). A full proof might be to hard (as you said), but I think that the core idea of repeated bisection can be explained pretty nicely with images. Point out where continuity is used.
Demonstrate that the IVT is very important because it is used to prove some very obvious results which are still not trivial to show without it, e.g.


*

*If you have a (convex) shape and a curve starting on the inside and ending on the outside of this shape, then this curve must have crossed the shapes boundary somewhere.


Or some more surprising results like


*

*There are two points on exactly opposite sides of the earth which have the exact same temperature and air pressure at the same time.


The proofs might be somehow hand wavy and not completely rigorous (especially in the last example), but you can give an idea on how to apply the theorem.
You can mention that the IVT is a very nice tool to show the existence of results without computing them, i.e. it is a (seemingly) non-constructive tool. It can be used to show that certain solutions or intersection points exist but do not bother with computing them if you are not interested in the exact value anyway.
A: Make the proof accessible to a large audience, by explaining technicalities in layman's terms. Most of the required concepts are intuitive. Show how intuition can be formalized.
