# Intermediate Value Theorem

## Theorem

Let $f: S \to \R$ be a real function on some subset $S$ of $\R$.

Let $I \subseteq S$ be a real interval.

Let $f: I \to \R$ be continuous on $I$.

Let $a, b \in I$.

Let $k \in \R$ lie between $f \left({a}\right)$ and $f \left({b}\right)$.

That is, either:

- $f \left({a}\right) < k < f \left({b}\right)$

or:

- $f \left({b}\right) < k < f \left({a}\right)$

Then $\exists c \in \left({a \,.\,.\, b}\right)$ such that $f \left({c}\right) = k$.

## Corollary

Let $0 \in \R$ lie between $f \left({a}\right)$ and $f \left({b}\right)$.

That is, either:

- $f \left({a}\right) < 0 < f \left({b}\right)$

or:

- $f \left({b}\right) < 0 < f \left({a}\right)$

Then $f$ has a root in $\left({a \,.\,.\, b}\right)$.

## Proof

This theorem is a restatement of Image of Interval by Continuous Function is Interval.

From Image of Interval by Continuous Function is Interval, the image of $\left({a \,.\,.\, b}\right)$ under $f$ is also a real interval (but not necessarily open).

Thus if $k$ lies between $f \left({a}\right)$ and $f \left({b}\right)$, it must be the case that:

- $k \in \operatorname{Im} \left({\left({a \,.\,.\, b}\right)}\right)$

The result follows.

$\blacksquare$

## Also known as

This result is also known as **Bolzano's theorem**, for Bernhard Bolzano.

Some sources attribute it to Karl Weierstrass, and call it the **Weierstrass Intermediate Value Theorem**.

## Also see

- Intermediate Value Theorem (Topology), of which this is a corollary

## Historical Note

This result rigorously proves the intuitive truth that:

*if a continuous real function defined on an interval is sometimes positive and sometimes negative, then it must have the value $0$ at some point.*

Bernhard Bolzano was the first to provide this proof in $1817$, but because of incomplete understanding of the nature of the real numbers it was not completely satisfactory.

Hence many sources refer to this as **Bolzano's Theorem**.

The first completely successful proof was provided by Karl Weierstrass, hence its soubriquet the Weierstrass Intermediate Value Theorem.

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 9.10$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($1815$ – $1897$)