On a sum involving the integral part function

Let [t] be the integral part of a real number t, and let f be the arithmetic function satisfying some simple condition. We establish a new asymptotical formula for the sum \(S_f (x)=\sum_{n\leqslant x}f([ x/ n ])\) $$S_f(x)=\sum _{n\le x}f([x/n])$$ , which improves the recent result of J. Stucky (2022).