What does the intermediate value theorem mean?
It means that a if a continuous function (on an interval
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!
You could say that it basically says the Real numbers have no gaps.
The intermediate value theorem states that if
In particular Bolzano's theorem says that if
Consider the function
This is a Real valued function which is continuous on the interval (in fact continuous everywhere).
We find that
This value of
So if we were considering
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.