Germ of a smooth function

Summary

The germ of a smooth function at a point $p$ of a smooth manifold is the function seen only on arbitrarily small neighbourhoods of $p$. It is the special case of a germ for smooth real functions. The germs at $p$ form a local ring, 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.