Let be a function that maps closed and bounded intervals onto closed and bounded intervals, hence if , then for some real numbers .
1)Prove that there is an contained in such that for all contained in
2)Prove the range of has the intermediate value propery.
3)Give an example showing not needing to be continuous.
My attempts/observations...
1) It would appear that we are looking for an upper bound for ? I'm not reall too sure how to start that one though...
2) Well, possibly we can define and and then use that for , . Then show that is between ???
3)I think I can probably take a better guess once I have a better understand of 1) and 2)
Any help would be greatly appreciated.
Ok... How about this...
1) Since then we have that is an upper bound. Thus, by definition, for any ,
2) I was just saying that a theorem in my notes states that if some function is continuous then the range of has the intermediate value property so I thought we could maybe use that.
But... in order for the range of to have the intermediate value property... here is my guess. For some where , , then there must be a such that for some . But the more I look at what I just said I'm guessing this only works for strictly increasing continuous functions so I'm sure that's not right.
The only definition that I have for the IVP is the following:
Let be continuous on some interval . If there are , such that and , then there exists a , such that .
This is the only defintion that I have. My professor didn't explicitly define the IVP for the range of a function.