I already proved that $f$ is of class $C^1$, but I have many doubts with this example:
- Why the collection $\{\phi_i\}$ forms a partition of unity on $\mathbb{R}$? I know that $\phi_i\geq 0$ for all $i=1,2,3,...$, plus $S_i$ is contained in $\mathbb{R}$, but Why $\sum_{i=1}^{\infty}\phi_i=1$? P?
I know that $\sum_{i=1}^{\infty}\phi_i=\sum_{m=0}^{\infty}f(x-m\pi)+\sum_{m=1}^{\infty}f(x+m\pi)$ but what else can I do? Thank you very much.
Edit: We know that $\phi_{2m}(x)=0$ for $|x+m\pi|>\pi$ and $\phi_{2m+1}(x)=0$ for $|x-m\pi|>\pi$ by definition, that is $\phi_{2m}(x)=0$ for $x\notin [-(m+1)\pi,(-m+1)\pi]$ and $\phi_{2m+1}(x)=0$ for $x\notin [(m-1)\pi, (m+1)\pi]$.
Let $x\in \mathbb{R}$, then there is a $m\in \mathbb{Z}$ such that $x\in [-(m+1)\pi,(-m+1)\pi]$ or $x\in [(m-1)\pi, (m+1)\pi]$, in the first case
$\sum_{i=1}^{\infty}\phi_i(x)=\phi_{2m}(x)+\phi_{2m+2}=1$ or
$\sum_{i=1}^{\infty}\phi_i(x)=\phi_{2m}(x)+\phi_{2m-2}=1$
In the same way the other case is made, is this reasoning correct?