Let $S=\{x_1,\cdots, x_N\}$ be a finite sequence of real numbers. I am interested in characterizing the family of functions $F$ such that for any $f\in F$ the set of numbers $$ c_\lambda =\sum_{i=1}^{N}f(x_i-\lambda),\;\lambda \in \mathbb{R} $$ uniquely identifies the sequence $S$ up to element permutations. My intuition is that, for example, any monotonic (not constant) $f$ will work (true?). Any suggestion how to characterize $F$ even partially?

Thanks!

Fabio