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

You'll learnthe two things the Intermediate Value Theorem needs and the one thing it promises, and what follows when a hypothesis fails, which is nothing at all in either direction.

A pen that never leaves the paper cannot get from one height to another without passing through every height in between. That is the whole theorem, and it buys exactly one thing: at least one input somewhere inside the interval. Not how many, not where, and nothing about a maximum. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · CONTINUOUS ON A CLOSED INTERVAL, SO AT LEAST ONE c · v2

continuous on a closed interval means every height between the end values is hit at least once

Before you start

What you're looking at

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

The question

If an unbroken curve starts below the line and finishes above it, what has been established, and what has not?

Step 1 / 12