Exercise 8.5.5

$h$ is continuous on the compact set $[-\pi ,\pi ]$ and therefore uniformly continuous over this set, and thus for any $𝜖>0$ we can find $\delta$ so that $\left|x-x_0\right|<\delta$ implies $\left|h(x)-h(x_0)\right|<𝜖∕2$, at least if $x$ and $x_0$ are both in $[-\pi ,\pi ]$ or both in the same “copy” of $h$. If they are not, however, then there must be some $k=\mathit{n\pi }$, with $n$ odd, separating them, with $\left|x-k\right|<\delta$ and $\left|k-x_0\right|<\delta$. Then

showing $h$ is uniformly continuous on all $R$.