# discussion

A germ is a mathematical concept that gives us a way to talk about when two functions are locally the same.

Roughly speaking, the germ of a function $s$ is the set of all other nearby functions that ‘‘look like’’ $s$ when you zoom in (along with the information of where these nearby functions are defined).

More formally, the germ of a function $s: V \to \mathbb{R}$ at a point $p \in V$ is the set of all open sets $U \subseteq V$ and smooth functions $f: U \to \mathbb{R}$ such that we can find an open set $W$ contained in both $V$ and $U$ that also contains $p$ such that the images of $s$ and $f$ agree when restricted to this $W$.

This is sometimes notated $\text{germ}_s(p)$, so the text says, ‘‘don’t spread $\text{germ}_s(p)$.’’