---
tag: 7SWSRGDT4A
kind: concept
name: Germ of a smooth function
msc: [58A05, 58K40, "26E10"]
specializes: [VZXF7RNFKS]
sources:
  - url: https://en.wikipedia.org/wiki/Germ_(mathematics)
---

## Summary

The germ of a [[smooth function|0V33BNKZ9S]] at a point $p$ of a [[smooth manifold|F45VQ47GPV]] is the function seen only on arbitrarily small neighbourhoods of $p$.
It is the special case of a [[germ|VZXF7RNFKS]] for smooth real functions.
The germs at $p$ form a [[local ring|MZ02BKPHJA]], and this ring carries all local information of smooth calculus at $p$.
Tangent vectors, jets and the classification of singularities are all built on germs.

## Explanations

### L1 human

Two smooth functions have the same germ at $p$ when they agree near $p$, however far apart they are elsewhere.
Think of a germ as what a function looks like under a microscope pointed at $p$.
For example, on $\R$ the function $f(x) = x$ and a smooth function $g$ that equals $x$ on $(-1, 1)$ and equals $0$ for $\abs{x} > 2$ have the same germ at $0$.
Germs can be added and multiplied, just like functions.
A germ is invertible exactly when its value at $p$ is not zero.