ZA-SAMO 2006 Problem 6

Consider the function $f$ defined by $$f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )$$ for all positive integers $n$ . (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ .) Prove that

(a) $f(n+1)>f(n)$ for infinitely many $n$ .

(b) $f(n+1)<f(n)$ for infinitely many $n$ .