# 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 $I$.

Then $f$ is a Darboux function.

That is:

Let $a, b \in I$.

Let $k \in \R$ lie between $\map f a$ and $\map f b$.

That is, either:

- $\map f a < k < \map f b$

or:

- $\map f b < k < \map f a$

Then $\exists c \in \openint a b$ such that $\map f c = k$.

## 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 $\openint a b$ under $f$ is also a real interval (but not necessarily open).

Thus if $k$ lies between $\map f a$ and $\map f b$, it must be the case that:

- $k \in \Img {\openint a b}$

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

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.4$: Continuity: Theorem $1.4.5$ - 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 ($\text {1815}$ – $\text {1897}$) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Bolzano's Theorem** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**continuous function**(vi) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**intermediate value theorem**