Intermediate value theorem

The intermediate value theorem says that if a function, $f$, is continuous over a closed interval $[a,b]$, and is equal to $f(a)$ and $f(b)$ at either end of the interval, for any number, c, between $f(a)$ and $f(b)$, we can find an $x$ so that $f(x)=c$.

This means that if a continuous function's sign changes in an interval, we can find a root of the function in that interval. For example, if $f(1)=-1$ and $f(2)=2$, we can find an $x$ in the interval $[1,2]$ that is a root of this function, meaning that for this value of x, $f(x)=0$, if $f$ is continuous. This corollary is called Bolzano's theorem.