Germ

Remarks

One often restricts to continuous, smooth or holomorphic maps, and writes $[f]_x$ for the germ. The value $f(x)$ depends only on the germ, and so do derivatives of all orders at $x$ when $f$ is smooth. More generally, let $\mathcal{F}$ be a filter on a set $S$. Two maps on $S$ have the same germ along $\mathcal{F}$ if they agree on some member of $\mathcal{F}$. The case above uses the neighbourhood filter of $x$.

Example: on $\R$, the maps $f(t) = t$ and $g(t) = \abs{t}$ have the same germ at $1$, since they agree on $W = (0, 2)$.

Non-example: on $\R$, let $f = 0$ and let $g(t) = e^{-1/t^2}$ for $t \ne 0$, with $g(0) = 0$. Both are smooth, and all their derivatives agree at $0$. Yet $g(t) \ne 0$ for every $t \ne 0$, so they agree on no neighbourhood of $0$, and their germs at $0$ differ.