I am having trouble understanding a proof to establish a specific version of Taylor's formula.
I'll first give the statement and then below cite the part where I am stuck, so here is what I'd like to prove:
Let m be a non-negative integer, $f \in S(\mathbb{R})$ (Schwartz Space) and $y \in \mathbb{R}$ be a fixed point. If $f$ and all its derivatives up to the order $m$ vanish at $y$ then there exist functions $h_\beta \in S(\mathbb{R})$ such that \begin{equation} f(x) = \sum_{\beta \colon\\, |\beta| = m + 1} (x-y)^\beta h_\beta(x), \qquad \forall x \in \mathbb{R} \end{equation}
And this is how the proof starts:
Let $\zeta \in C^\infty_0(\mathbb{R}^n)$ and $\zeta \equiv 1$ in a neighborhood of the point $y$. Denote $f_1 = (1 - \zeta)f$ and $f_2 = \zeta f$. Obviously, the function \begin{equation} h(x) := |x-y|^{-2m-2}f_1(x) \end{equation} belongs to $S(\mathbb{R})$.
The last sentence is what troubles me - how is it obvious that the function above is Schwartz in $\mathbb{R}^n$? In particular, what is the value of $h(x)$ at $x = y$ ? I would say it's not defined, and even if the limit exists because $f_1(y) = 0$ I cannot have $h$ to be smooth then ...
What am I missing ? Thks alot for helping me!