This is problem 1.5.8 in Mathematical Statistics by Bickel and Doksum. It seems straightforward, but I am not sure if my proof is lacking in some way. It doesn't seem quite correct.
Question Let $X_1, X_2,..., X_n$ be a sample from some continuous distribution $F$ with density $f$, which is unknown. Treating $f$ as a parameter, show that the order statistics $X_{(1)},...X_{(n)}$ (cf. Problem B.2.8) are sufficient for $f$.
My Solution Attempt The sample density is simple the product of the $f(x_i)$. This yields
$$\prod_{i=1}^nf(X_{i})$$
We apply the factorization theorem where $h(x)=1$
$$g(\theta)=\prod_{i=1}^nf(X_{i})$$
(here $\theta=X_{(1)},...X_{(n)}$). And so because we have factorized the sample density, the statistics are sufficient.