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.