Intermediate Value Theorem

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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


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