Germ of a function

Definition

Definition for a function of one variable

Suppose $f$ is a function on a subset of $\mathbb{R}$ and $x_0$ is a point in the interior of the domain of $f$. The germ of $f$ is the collection of all functions $g$ defined on subsets of $\mathbb{R}$ containing $x_0$ in the interior of the domain, such that there exists an open subset $U\ni x_0$ with $U\subseteq \mathrm{d}\mathrm{o}\mathrm{m}f\cap \mathrm{d}\mathrm{o}\mathrm{m}g$. for which $f(x)=g(x)\mathrm{\forall }x\in U$. If $g$ is in this collection, we say that $f$ and $g$ have the same germ at $x_0$. The relation of having the same germ is an equivalence relation.

Intuitively, the germ of a function at a point describes how the function behaves very close to the point, where "very close" allows us to consider an arbitrarily small open subset containing the point. All "local" behavior at the point, including continuity, differentiability, and the values of the derivatives, depends only on the germ of the function at the point.