So I'm trying to find a solution to this question. In part (a), I have shown that for a function which is continuous on satisfying then for any number there is some for which I did this by considering the function and assuming for any We can then show that either or by noting that, since is continuous we must have or for all So, in the latter case for example, we have

This clearly relies on the fact that with

Now for part (b), which has me totally stumped. For with for any find a function continuous on satisfying for which for any I just can't see how that could work. Imagine any function, and connect the points and with a straight line - surely if we have this line must be horizontal at some point!

How do I find such an ?