In this question the OP asks for a solution for the problem:
Find all $f(x)$ satisfying $ f(f(x)) = x^2-2$.
One answer squarely states, that no solution exists (and even gives a reference to a jstor-accepted article which states this as a general theorem on this type of problem), and in another answer someone else (in this case I) happen to show one solution.
So - being an amateur - I first assumed that something with my solution was wrong (but had no idea about what this could be). After that, I suspected that the general theorem might be only for a certain sub-class of that type of problems, but I cannot find a key for that in the article; on the contrary, the authors emphasize that indeed no solution of whatever form can exist.
Now that question/answers are of 2013, so of nearly 2 years ago and the two statements still stay parallel in the same question, stating opposite "truths".
How can I / we resolve that problem? Put a comment below of each of the controversy statements? Put a decisive comment below the question?
Shall we leave the contradicting answers as they are (which, in my view is a very uneasy situation for any casual reader without own deeper insight) or should some comment be made, possibly at each of the contradictory answers, or possibly the authors be pinged to add some explanations into their answers in the view of the concurring statements about the existence of a solution?
[added some information]
Here is a bit from the article:
THEOREM 1. Let $P$ be a polynomial of degree 2 defined on the entire complex plane C. Then P has no iterative roots of any order whatever; i.e., for any integer $r \ge 2$, there exists no function $f$ whatever mapping C into itself such that $f°^r= P$.
It must be emphasized that the phrase "no function f whatever" is to be taken literally and does not mean merely "no entire function" or "no continuous function," etc. From this it is to be expected that none of the usual methods of analysis or topology play a role in what follows. This is the case. The proof of Theorem 1 is, in essence, purely combinatorial.
I assume this is the part which leads to the introduction in the answer which refers to that article:
There is no such f