  Let $f$ be an arithmetic function satisfying some simple conditions. The aim
of this paper is to establish an asymptotical formula for the quantity \[
S_f(x):=\sum_{n\leq x}\frac{f([x/n])}{[x/n]} \] as $x\rightarrow\infty$, where
$[t]$ is the integral part of the real number $t$. This generalizes some recent
results of Bordell\`es, Dai, Heyman, Pan and Shparlinski.
