A difficult exercise

I must confess that I did not know how to solve this exercise:

Let $f \colon \mathbb{R}^{+} \to \mathbb{R}^{+}$ be a continuous function. Prove that $\lim_{x \to +\infty} f(x)=0$ if and only if $\lim_{n \to +\infty} f(n t)=0$ for every $t>0$.

A solution can be found here.

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