# The Intermediate Value Theorem Functions that are continuous over intervals of the form $[a,b]$, where $a$ and $b$ are real numbers, exhibit many useful properties. Throughout our study of calculus, we will encounter many powerful theorems concerning such functions. The first of these theorems is the Intermediate Value Theorem: > [!info]+ The Intermediate Value Theorem > If $f$ is a **continuous** function over a closed, bounded interval $[a,b]$. > > If $z$ is any real number between $f(x)$ and $f(b)$, then there is a number $c$ within the interval $[a,b]$ that satisfies the function $f(c)=z$. > > ![[Pasted image 20220517181708.png]] > [!question] Practice: Application of the Intermediate Value Theorem > [PENDING](https://openstax.org/books/calculus-volume-1/pages/2-4-continuity)