Germ

Summary

The germ of a function at a point records the behaviour of the function on arbitrarily small neighbourhoods of the point and forgets the rest. Two functions have the same germ at $x$ when they agree on some neighbourhood of $x$. Germs make local properties, such as derivatives and local power series, into properties of single objects. They are the stalks of the sheaf of functions and appear throughout analysis, geometry and singularity theory.