Suppose $f$ is some real function with the above property, i.e.
if $\sum\limits_{n = 0}^\infty {x_n}$ converges, then $\sum\limits_{n = 0}^\infty {f(x_n)}$ also converges.
My question is: can anything interesting be said regarding the behavior of such a function close to $0$, other than the fact that $f(0)=0$?