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## Main Question or Discussion Point

Let ##X## be the set of all functions infinitely differentiable at ##0##. Let's define an equivalence relation on $X$ by saying that ##f\sim g## if there exists a sufficiently small open interval ##I## containing ##0## such that ##f(x)=g(x)## for all ##x## in ##I##. Then the set of germs of ##X## denotes the set ##Y## of equivalence classes of elements of ##X## under this equivalence relation.

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.

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.