# Difference between revisions of "Intermediate Value Theorem"

The Intermediate Value Theorem is one of the very interesting properties of continous functions.

## Statement

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $f$ be continous on $[a,b]$

Let $f(a)

Then, $\exists c\in (a,b)$ such that $f(c)=k$

## Proof

Consider $g:[a,b]\rightarrow\mathbb{R}$ such that $g(x)=f(x)-k$

note that $g(a)<0$ and $g(b)>0$

By Location of roots theorem, $\exists c\in (a,b)$ such that $g(c)=0$

or $f(c)=k$

QED