Let be continuous with andAssume f is injective but not decreasing. This implies f is either an oscillating function, strictly increasing, increasing or constant.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 .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! ).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
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!