Intermediate Value Theorem

Continuous, with values straddling N, so some c has f(c) = N. Existence of roots, made easy.

The idea

Theorem (Intermediate Value Theorem).

Let $f$ be continuous on $[a, b]$, and let $N$ be any number lying between $f(a)$ and $f(b)$. Then there is some $c$ in $(a, b)$ with $f(c) = N$.

The claim is that a continuous function cannot skip values: to travel from the height $f(a)$ to the height $f(b)$, an unbroken graph must cross every height in between. Continuity is the essential hypothesis — a function permitted to jump can pass over $N$ without ever taking it.

The usual application is proving that an equation has a solution. If $f$ is continuous and $f(a)$ and $f(b)$ have opposite signs, then $0$ lies between them, so $f$ has a root in $(a, b)$. This establishes the existence of roots for equations that algebra cannot solve.

Note what is not claimed. The theorem produces no $c$ and narrows down no location; it promises at least one such point, does not say how many, and says nothing about how $f$ behaves in between. It is an existence statement only.

Ways to work on it

• Walkthrough. State the Intermediate Value Theorem, then use it to locate a root of a cubic.
• Practice. Show a polynomial has a root by finding a sign change.
• Hardest. Conceptual: when does the IVT apply, and what does it (not) tell you?

Not sure where to start? Take the ten-question placement test.