# What does the intermediate value theorem mean?

Jan 5, 2016

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 $f \left(b\right)$.

#### Explanation:

In order to remember or understand it better, please know that the math vocabulary uses a lot of images. For instance, you can perfectly imagine an increasing function! It's the same here, with intermediate you can imagine something between 2 other things if you know what I mean. Don't hesitate to ask any questions if it's not clear!

Jan 5, 2016

You could say that it basically says the Real numbers have no gaps.

#### Explanation:

The intermediate value theorem states that if $f \left(x\right)$ is a Real valued function that is continuous on an interval $\left[a , b\right]$ and $y$ is a value between $f \left(a\right)$ and $f \left(b\right)$ then there is some $x \in \left[a , b\right]$ such that $f \left(x\right) = y$.

In particular Bolzano's theorem says that if $f \left(x\right)$ is a Real valued function which is continuous on the interval $\left[a , b\right]$ and $f \left(a\right)$ and $f \left(b\right)$ are of different signs, then there is some $x \in \left[a , b\right]$ such that $f \left(x\right) = 0$.

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Consider the function $f \left(x\right) = {x}^{2} - 2$ and the interval $\left[0 , 2\right]$.

This is a Real valued function which is continuous on the interval (in fact continuous everywhere).

We find that $f \left(0\right) = - 2$ and $f \left(2\right) = 2$, so by the intermediate value theorem (or the more specific Bolzano's Theorem), there is some value of $x \in \left[0 , 2\right]$ such that $f \left(x\right) = 0$.

This value of $x$ is $\sqrt{2}$.

So if we were considering $f \left(x\right)$ as a rational valued function of rational numbers then the intermediate value theorem would not hold, since $\sqrt{2}$ is not rational, so is not in the rational interval $\left[0 , 2\right] \cap \mathbb{Q}$. To put it another way, the rational numbers $\mathbb{Q}$ have a gap at $\sqrt{2}$.

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The big thing is that the intermediate value theorem holds for any continuous Real valued function. That is there are no gaps in the Real numbers.