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Some functions, like say f(x)=x, have a limited number of derivatives that can be taken before non-unique results are obtained past a certain order i.e. the 2nd derivative of x wrt to x = 0 and the 3rd derivative is 0 and so on. Is there an existing function w/ analogous results for integrals?

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No there isn’t, for the reason you’ve just stated.

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