Let $X$ be a metric space.Consider the set $A=\{(U,f): U $open set containing $p$,$f:U\to \mathbb R$ continuous$\}$.Define an equivalence relation on $A$ by $(U,f)\sim (V,g)$ if $\exists$ an open set $W\subset U\cap V$ such that $p\in W$ and $f(x)=g(x)$ for all $x\in W$.
Then equivalence classes of $\sim$ are called germs at $p$.Consider the set of all germs at $p$ and define $+$ and $.$ on $A/\sim$ by $[(U,f)]+[(V,g)]=[U\cap V,f+g)]$ and $[(U,f)].[(V,g)]=[(U\cap V,fg)]$.Then $A/\sim$ forms a ring .Now we want to look at units of this ring.I think the units of this ring are $[(U,f)]$ such that $f(p)\neq 0$.Is it correct?What special can we say about this ring?