Question1. Let where . Find all (kinds of) functions such that for all in some (nice) interval, (we will now call this odd function) where is continuous and not an odd function? (For the sake of sanity, let us exclude cases like the following: where is some odd function and is constant. I hope we are on the same page here about the interesting cases. Can you find other interesting examples? Maybe some mathematical ingenuity and creativity comes into play?)
Here are some well-known examples for :
(Note: Let us exclude the case where we define as the inverse function of or vice versa, ie )
Case1: being a logarithmic function and so that we can use the identity
(in fact, the third example is the only example I can find which involves two transcendal functions.)
Case2: (Any peculiarities?)
(1) Define as the n-th iterate of . (-> FALSE STATEMENT)
(2) There always exists some for any odd function , where is not an odd function.
Question3. Using the function from Q2, find a solution such that for all n, is not odd BUT is odd, where is not odd.
LEAVE A NOTE FOR THE FOLLOWING: Errors, better notations, more examples, anything helpful, etc