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PF Gold
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 Quote by Identity Sorry, I've never heard of the term 'germ' before, can you explain please?
I assumed that by $C^{\infty}(p)$ you mean the set of real-valued 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 $C^{\infty}(p)$ are called germs of smooth functions.