Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

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.

Show more
LEARN MORE EFFECTIVELY AND GET BETTER GRADES!
Ask a Question