We were given this information:
"A set $U \subseteq \mathbb{R}^n$ is open if for every $p \in U$ there exists an $\epsilon =\epsilon(p) > 0$ so that for every $y \in \mathbb{R}^n$ such that $\lVert y-x \rVert <0$ we have $y \in U$.
Let's say that $F: \mathbb{R}^n \to \mathbb{R}$ is differentiable at $p$ if there exists $\vec{a}, \vec{b} \in \mathbb{R}^n$ such that $$\lim_{x \to p} \frac{f(x)-(\vec{a}\cdot x+b)}{\lVert x-p \rVert}.$$
The sheaf of differentiable functions refers to the canonical identification, $U \mapsto C^1(U)$ which identifies to each open set, $U \in \mathbb{R}^n$, the set of differentiable functions on that set, $C^1(U) := \lbrace f:U\to \mathbb{R} |f \text{ differentiable}\rbrace$. For a given point $p$, one can define an equivalence relation on $C(\mathbb{R^n})$ by $$f \sim g \text{ if there exists an open set } U \subset \mathbb{R^n} \text{so that} f\vert _U = g\vert _U.$$
The set of equivalence classes is referred to as the stalk over p and is denoted $\mathcal{O}(p)$.
Show that the stalk, $\mathcal{O(p)}$, is a vector space by proposing a well-defined addition and scaling operation, $\lambda [f] + [g] := [\lambda f + g]$."
I think I'm just not very clear on how to prove something is well defined, or even how to begin to consider this prove.
In order to prove that addition is well defined, all you need to do is show independence of choice of representatives. That is, show that for differentiable functions $f_1,f_2,g_1,g_2$, defined around $p$, if $f_1\sim g_1$ and $f_2\sim g_2$, then $f_1+f_2\sim g_1+g_2$. For scalar multiplication you also need to perform a similar check.