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 (*unordered*) sequence $S$ (for example $f=(1/N)H$ where $H$ is the Heaviside function works).
My intuition is that, for example, any monotonic (not constant) $f$ will work (true?). Any suggestion how to characterize $F$ even partially?

Thanks!

Fabio