An Existence Theorem, and Four Ways to Ask It for More

You'll learnto state the two hypotheses of the Intermediate Value Theorem and check both before invoking it, to read its conclusion as bare existence rather than as a count or a location, and to say what follows when the hypotheses fail — which is nothing at all, in either direction.

The Intermediate Value Theorem is the formal version of a fact about drawing: a pen that never leaves the paper cannot get from one height to another without passing through every height in between. That is the entire content, and it buys exactly one thing — the existence of at least one input somewhere inside the interval. It does not say how many there are, it does not say where any of them is, and it says nothing whatever about maxima. Almost every mark lost on this topic is lost by asking the conclusion for a second sentence it never had.

8 STEPS · 6 QUICK CHECKS · CONTINUOUS ON A CLOSED INTERVAL ⇒ AT LEAST ONE c · v1

continuous on a closed interval ⇒ every height between the endpoint values is taken at least once

Before you start

What you're looking at

One graph. A horizontal line marks the target height N. Two filled discs mark the values the function takes at the two ends of the closed interval, and the shaded band between them is every height the theorem talks about.

The question

If the curve starts below the target line and finishes above it without ever breaking, what exactly has been established — and what has not?

Watch for

Every emerald disc sitting on the target line is a valid answer. Several steps mark a point in red instead: those are claims the picture refuses, and each one is a claim the theorem was never making.

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