I went back to read some manifold theory recently and I realized that I can't justify to myself the reason to consider germs of smooth functions over simply smooth functions other than formalism, because we know that a point-derivation of smooth functions depends only on the germ of a given smooth function, that is, its local behaviour around a point. But for germs we can always consider a representative defined on the whole manifold via bump functions, and the algebraic structure of the space of germs $\mathcal{C}^{\infty}_p (\mathcal{M}) $ is nothing more special than the ring structure of $\mathcal{C}^{\infty}(\mathcal{M})$, so I have this question: what do I earn by considering germs of functions, really?


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