Let $a \in \mathbb{R}$ be a point and $S=\mathbb{R}[x]_{(x-a)}$ the localization of the polynomial ring $\mathbb{R}[x]$ with maximal ideal $(x-a)$.
i) Describe the elements of $S$ as differentiable functions on $\mathbb{R}$, defined on a neighbourhood of $a$ on $\mathbb{R}$.
ii) Show that there is not one neighbourhood for all functions in $S$.
I'm really unsure how to tackle this one. For i) I'd take the function $f:[a-\varepsilon, a+\varepsilon] \rightarrow \mathbb{R}$ with $f(x)=\frac{p(x)}{(x-a)}$ where $p(x)$ is a polynomial in $\mathbb{R}[x]$ and $f(x)=0$ for $x=a$. Does that make sense?
For ii) I don't know exactly how I might prove that. Any advice?
Thanks.