Let the function $f\colon \mathbb{R} \to\mathbb{R}$ have arbitrarily small positive periods, in the sense that if $\delta>0$ then there exists $T \in (0,\delta)$ such that $f(t + T) = f(t)$ for all $t \in \mathbb{R}$.
(i) Prove that if $f$ is continuous then $f$ is constant.
(ii) What if $f$ is not assumed to be continuous?