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What uniquely characterizes the germ of a smooth function?
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[QUOTE="mathwonk, post: 6051996, member: 13785"] i suspect what mathman meant was that the sequence of derivatives of all orders at 0, is the minimum amount of identifying information of a germ, for functions that are analytic, and presumably only for those. In general I do not believe there is any minimal identifier. I.e. the only data that will determine a germ is the values of the function on some interval, but the values on any smaller interval will also work, so no interval is minimal. well ok, since the function is continuous, knowing its values on any dense subset of an interval will do. again, not only is the interval not minimal, the dense subset isn't either. so i do not know of any useful solution to this problem. in the analytic case of course the germs are in one one correspondence with the convergent power series at zero. and in the algebraic case, the germs are in one one correspondence with the "local ring" at the origin, i.e. quotients of polynomials, with bottoms having a non zero constant term.but since the question to describing the quotient space of all germs, maybe there is a way, and maybe andrewkirk's idea to also throw out the known subspace of analytic stuff may help, but i don't see how to describe it. it may be illuminating to read the sections in courant's calculus referred to as "order of vanishing" in his index. there he discusses the complexity of the related problem of comparing how fast various functions approach zero or infinity as x-->0 or infinity. essentially there is no way to assign a number to this even if we use all real numbers. in particular there are functions that approach infinity faster than x but slower than x^(1+e) for all e > 0. thus no real number exists that measures the speed of increase of such functions. (courant, vol. 1, p. 193). similarly there are functions that approach zero more slowly and others that approach more rapidly than any power of x. [/QUOTE]
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What uniquely characterizes the germ of a smooth function?
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