Boundedness of Continuous and Periodic Functions on $\mathbb{R}$

Let $T>0$ be a period of $f$. On $[0,T]$, $f$ is bounded by some $M$ because $f$ is continuous on a closed and bounded interval. For any $x\in \mathbb{R}$, $x−nT\in[0,T]$ for $n=\lfloor x/T\rfloor$, so $∣f(x)∣=∣f(x−nT)∣≤M$. Thus, $f$ is bounded by $M$ on $\mathbb{R}$.

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Fixed Point Theorem for Continuous Functions on a Closed Interval

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