# Intermediate Value Theorem

## Key Questions

• To answer this question, we need to know what the intermediate value theorem says.

The theorem basically sates that:
For a given continuous function $f \left(x\right)$ in a given interval $\left[a , b\right]$, for some $y$ between $f \left(a\right)$ and $f \left(b\right)$, there is a value $c$ in the interval to which $f \left(c\right) = y$.

It's application to determining whether there is a solution in an interval is to test it's upper and lower bound.

Let's say that our $f \left(x\right)$ is such that $f \left(x\right) = {x}^{2} - 6 \cdot x + 8$ and we want to know if there is a solution between $1$ and $3$ (in the $\left[1 , 3\right]$ interval).
$f \left(1\right) = 3$
$f \left(3\right) = - 1$
From the theorem (since all polynomials are continuous), we know that there is a $c$ in $\left[1 , 3\right]$ such that $f \left(c\right) = 0$ ($- 1 \le 0 \le 3$)//

Hope it helps.

There are several definitions of continuous function, so I give you several...

#### Explanation:

Very roughly speaking, a continuous function is one whose graph can be drawn without lifting your pen from the paper. It has no discontinuities (jumps).

Much more formally:

If $A \subseteq \mathbb{R}$ then $f \left(x\right) : A \to \mathbb{R}$ is continuous iff

$\forall x \in A , \delta \in \mathbb{R} , \delta > 0 , \exists \epsilon \in \mathbb{R} , \epsilon > 0 :$

$\forall {x}_{1} \in \left(x - \epsilon , x + \epsilon\right) \cap A , f \left({x}_{1}\right) \in \left(f \left(x\right) - \delta , f \left(x\right) + \delta\right)$

That's rather a mouthful, but basically means that $f \left(x\right)$ does not suddenly jump in value.

Here's another definition:

If $A$ and $B$ are any sets with a definition of open subsets, then $f : A \to B$ is continuous iff the pre-image of any open subset of $B$ is an open subset of $A$.

That is if ${B}_{1} \subseteq B$ is an open subset of $B$ and ${A}_{1} = \left\{a \in A : f \left(a\right) \in {B}_{1}\right\}$, then ${A}_{1}$ is an open subset of $A$.

It means that a if a continuous function (on an interval $A$) takes 2 distincts values $f \left(a\right)$ and $f \left(b\right)$ ($a , b \in A$ of course), then it will take all the values between $f \left(a\right)$ and $f \left(b\right)$.