Putnam 1964 B3

3. Let $f(x)$ be a real continuous function defined for all real $x$. Assume that for every $\epsilon>0$ $$\lim _{n \rightarrow \infty} f(n \epsilon)=0, \text { (where } n \text { is a positive integer) }$$

Prove that $$\lim _{x \rightarrow \infty} f(x)=0$$