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# What does the intermediate value theorem mean?

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

The 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.

color(white)() 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).

color(white)() 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.