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?
(DISCLAIMER: This discussion is NOT about finding all possible solutions but brainstorming as much examples as possbile. 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 Also, where is some odd function and is constant.)
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: (Peculiar stuff) (NOT INTERESTING)
(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