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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?

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    Pippo, what methods have you tried for this problem? For part (ii), consider using arbitrarily small periods "T" that are rational numbers. Does it limit what "f" can be on the irrationals?2012-11-27

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