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Sorry, I've never heard of the term 'germ' before, can you explain please?

I assumed that by [itex]C^{\infty}(p)[/itex] you mean the set of realvalued functions f that are defined and smooth on some neighborhood U of p, modulo the equivalence relations according to which f~g iff f and g coincide on some small nbhd of p.
If so, then the elements of [itex]C^{\infty}(p)[/itex] are called germs of smooth functions.