If $x$ is a point on a smooth manifold $M$ , then a germ of smooth functions near $x$ is represented by a pair $(U,f)$ where $U \subseteq M$ is an open neighbourhood of $x$ , and $f$ is a smooth function $U \rightarrow \mathbb{R}$ . Two such pairs $(U,f)$ and $(V,g)$ are considered equivalent if there is a third open neighbourhood $W$ of $x$ , contained in both $U$ and $V$ , such that $f|_W=g|_W$ . To be precise, a germ of smooth functions near $x$ is an equivalence class of such pairs.

In more fancy language: the set $\mathcal{O}_x$ of germs at $x$ is the stalk at $x$ of the sheaf $\mathcal{O}$ of smooth functions on $M$ . It is clearly an $\mathbb{R}$ -algebra.

Germs are useful for defining the tangent space $T_x M$ in a coordinate-free manner: it is simply the space of all $\mathbb{R}$ -linear maps $X:\mathcal{O}_x \rightarrow \mathbb{R}$ satisfying Leibniz' rule $X(fg)=X(f)g+fX(g)$ . (Such a map is called an $\mathbb{R}$ -linear derivation of $\mathcal{O}_x$ with values in $\mathbb{R}$ .)