Let be continuous with and
Assume f is injective but not decreasing. This implies f is either an oscillating function, strictly increasing, increasing or constant.Quote:
i). If also f is injective prove that f is decreasing.
f cannot be an oscillating function because if f was an oscillating function over then if where and then by the definition of an oscillating function such that . This would contradict the fact that f is injective.
So we now know that f is either strictly increasing, increasing or constant.
Once again, f cannot be a constant function. If f was a constant function (similar to before) if then .
f is either a strictly increasing or an increasing function.
If f is strictly increasing then if then .
This gives a contradiction since so f cannot be strictly increasing.
A similar argument holds for an increasing function:
If f is increasing then if then .
This contradicts the definition of injectivity since it is possible for .
Hence we are forced to conclude that f is a decreasing function.
If f only takes positive values then . Since f is also continous, f must be bounded since it's domain is bounded (general result). f is bounded above by and below by .Quote:
ii) if f takes only positive values show that there exists a positive real number with , carefully stating any general resullts you use.
Take for some .
This gives .
Hence as required.
This is the part i'm really having trouble with (provided my previous two proofs are okay! (Worried)).Quote:
b). Suppose is increasing and satisfies the conclusion of the intermediate value theorem. Prove that f is left continuous (ie. prove that the restriction of f to is continuous at c).
I need to find when where .
We now need to find when
...and this is where I get stuck (Crying)
Can someone please tell me whether my two above proofs are fine and show me how to do this last part?
I think there's a way of doing parts i) and ii) using the intermediate value theorem, but I couldn't see that way of doing it.
Any help is appreciated. Thankyou!