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My question, what uniquely characterizes the germ of a smooth function? That is to say, what is the minimum information needed to unambiguously specify a single element of ##Y## as opposed to all other elements of ##Y##? The nth derivatives of ##f## for all ##n## isn’t enough information, because the function f defined by ##f(x)=e^{-\frac{1}{x^2}}## when ##x## does not equal ##0## and ##f(0)=0## has the same nth derivatives as the function ##g(x)=0## for all ##n##, but they still don’t belong to the same germ.