If the function $g: R\rightarrow R$ is periodic, then $a_n=g(n)$ is periodic.

Question

If the function $g: R\rightarrow R$ is periodic, then $a_n=g(n)$ is periodic. Is the converse true? Give a proof or counterexample.

Answer 1

Both implications are false. For the statement itself consider @Bernard's comment: $sin(x)$ has trancendental period length, so that the period does not show in $\mathbb{N}$.

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As one of many counterexamples for the converse, consider the non-periodic function

$$f(x) = \begin{cases} 0 &\mbox{if } x \in \mathbb{N} \\ x & \mbox{else} \end{cases}$$

that becomes periodic (even constant) when restricted to $\mathbb{N}$.