What does the intermediate value theorem mean?

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!

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

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

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Consider the function `f(x) = x^2-2` and the interval `[0, 2]`.

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

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

This value of `x` is `sqrt(2)`.

So if we were considering `f(x)` 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 `[0, 2] nn QQ`. To put it another way, the rational numbers `QQ` 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.