I'm having trouble understanding the following argument (which I believe to be somewhat incomplete or flawed). Let $A=C(X)$ be the set of continuous functions from the topological space $X$ to the complex plane $\mathbb{C}$. We define $m_{x} = \{f \in C(X): f(x) = 0 \}$ and $A_x$ the ring of germs at point $x$. The statement is the following $A_x \simeq A_{m_x}$.
(1) I don't see how one defines $A_{m_x}$ since the set contains global functions that might not be well-defined as we quotient by functions $f$ such that $f(x) \neq 0$, but it doesn't necessarily mean that $f \neq 0$. Though it's indeed well defined in a neighborhood of $x$.
(2) Now using the universal property of localization, we sure want to define $\phi : A_{m_x} \rightarrow A_x$ s.t we have $\phi(a/s) = \iota(a)\iota(s)^{-1}$ where $\iota$ is the inclusion map $\iota: A \rightarrow A_x$. We want $\phi$ to be an isomorphism. It is surjective; now we want it to be one-on-one. Now I don't see how this is possible as $\phi(a/s) = 0$ iff $a = 0$ in a neighborhood of $x$, which doesn't imply that $a=0$ globally.
I guess there's something I don't really fathom, or my textbook might just be flawed. Anyway, thanks for your help.