\beginsection 18.8

Suppose that $f$ is a real-valued continuous function in $R$ and that
$f(a)f(b)<0$ for some $a,b\in R$.
Prove that there exists $x$ between $a$ and $b$ such that $f(x)=0$.

\medskip

From $f(a)f(b)<0$ we conclude that
$f(a)$ and $f(b)$ have opposite signs and neither is zero.
Therefore either $f(a)<0<f(b)$ or $f(b)<0<f(a)$.
In both cases we have the existence of $f(x)=0$ and $x\ne a,b$ by
the intermediate value theorem.