My question is as above.
Currently I am stuck at the very start, part (i)! I can't come up with an appropriate $f$, even though I've been thinking about it for ages! If someone would be able to give me a hint as to what sort of properties $f$ would have, so that I can then try to determine what $f$ should be, then I'd be most appreciative! (If I still can't do it, then perhaps someone can just tell me what $f$ should be!)
Once I've got $f$, I would hope to be able to continue with the rest of the question (hope is the key word!), so I currently am only looking for help for the first part.
Thanks in advance! :)
PS - Hopefully this isn't a duplicate question - I've had a brief look, but quite hard to search for something so specific as this!
Hint: Try $f_1(t) = t + g(t)$ as a starting point. That certainly has the algebraic property you want...but does it take a disk into itself? (I think that this means that there's a disk $D$ with $f(D) \subset D$, rather than $f(D) = D$, which would be another reasonable interpretation). Probably not.
To prove the disk=-mapping property, you want to use the stuff you know. You know that $\| g(t) - t \| \le K$. Hmm. So you know something about $g(t) - t$, but that doesn't tell you anything about $g(t) + t$, which is what you're looking at.
Can you think of a better function to use than $f_1$, knowing that you want to reason about $g(t) - t$?