germ of smooth functions

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 $U\to ℝ$. 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 ${𝒪}_x$ of germs at $x$ is the stalk at $x$ of the sheaf $𝒪$ of smooth functions on $M$. It is clearly an $ℝ$-algebra.

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