Sheaf of germs of continuous real valued functions on $\mathbb{R}$

Question

I encountered the phrase "sheaf of germs of continuous real-valued functions on $\mathbb{R}$" on Page 5, Example 1.4 of the book "Sheaf Theory" by Bredon.

I am trying to figure out the explicit description of this sheaf without using presheaf. Is there a way to do it? I have tried to formulate something (which I am stating below).

We know that a sheaf of abelian groups on a topological space $X$ is a pair $(\mathcal{A}, \pi)$ where:

1. $\mathcal{A}$ is a topological space.

2. $\pi$ is a local homeomorphism.

3. For each $x \in X$, $\mathcal{A}_x := \pi^{-1}(x)$ is an abelian group.

4. The group operations are continuous.

So for our case, we let $X := \mathbb{R}$ with the usual topology. Now what should be $\mathcal{A}$? Will it be $\mathcal{A} = C(X)$, the space of all continuous maps from $\mathbb{R}$ to $\mathbb{R}$? If yes, then what should be $\pi$?