Solution(s?) of the functional equation $f(2f(x)) = 2x$

Question

I want to prove that the only solution of $f(2f(X)) = 2X$ is $f(X) = X.$

I tried doing it like this:

• If $f(X) > X$ then $ f(2f(X)) > 2X$
• If $f(X) < X$ then $f(2f(X)) < 2x$
• Hence $f(X)$ must be equal to $X$.

I got told we cannot do this but didn't fully understand why not? Can somebody explain ??