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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 .

Let .

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!