Can we assume a function to be a polynomial in olympiads ??

Question

While solving Olympiad level functional equation questions, I have often found that assuming the functional equation to be a polynomial makes it easier to solve and in $99\%$ of the cases the function given in the solution is a simple polynomial.

So is it fine to assume the function to be a polynomial and proceed ??

Answer 1

If they ask you to find all functions, then no. You may say that if their proposed solution only involves polynomials, then you are fine, but you actually didn't prove that your solution is complete.