I may be asking something a little out of my comfort zone at this moment so bear with me.
Before I begin let me provide some background for the interested outsider:
Let $X$ be a topological space and fix some $x\in X$. We can consider the set $\mathcal{U}_x$ of all neighborhoods of $x$ in $X$. This is a directed set with the reverse inclusion ordering $\subseteq_{\text{op}}$. We can note then that if we consider the set $\{C(U,\mathbb{R})\}_{U\in\mathcal{U}_x}$ (where each $C(U,\mathbb{R})$ denotes, as usual, the set of continuous maps $U\to\mathbb{R}$ where we are thinking about it as a $\mathbb{R}$-space) and the set $\{\text{res}_{U,V}:C(U,\mathbb{R})\to C(V,\mathbb{R})\}_{U,V\in\mathcal{U}_x,\; U\subseteq_\text{op}V}$ (where $\text{res}_{U,V}:C(U,\mathbb{R})\to C(V,\mathbb{R})$ is the restriction map taking $C(U,\mathbb{R})\ni f\mapsto f_{\mid V}\in C(V,\mathbb{R})$) then we have a nice directed system of $\mathbb{R}$-spaces. We can then consider the direct limit $\varinjlim C(U;\mathbb{R})$ to get a $\mathbb{R}$-space. Call this space the germs of continuous functions at $x$ and denote it $\mathcal{G}_x$. It's not hard to prove (it's intuitively obvious from the direct limit definition) that $\mathcal{G}_x$ admits a different description as $C(X,\mathbb{R})/\sim$ where $f\sim g$ if and only if there exists $U\in\mathcal{U}_x$ with $f_{\mid U}=g_{\mid U}$ and we add/multiply scalars representative wise.
Anyways, so I completely understand (or at least I hope I do) the intuitive idea behind the germ of continuous functions. That said, I have a hard time actually computing it except for very extreme scenarios. For example, if $x$ is an isolated point of $X$ I see quite clearly that $\mathcal{G}_x\to\mathbb{R}:[f]\mapsto f(x)$ is a well-defined isomorphism. But, in general, I have no idea how to find nice descriptions of the germs of functions at a point. Is there any general methodology in finding them, or some general class of examples that admit a nice description?
Thanks for your time!
