We return to the question of the existence of a primitive function of f (x) = 1/x for x > 0 posed above. Since the function f(x) = 1/x is Lipschitz continuous on any given interval [a, b] with 0 < a < b, we know by the Fundamental Theorem that there is a unique function u(x) which satisfies u′(x) = 1/x for a ≤ x ≤ b and takes on a specific value at some point in [a, b], for example u(1) = 0. Since a > 0 may be chosen as small as we please and b as large as we please, we may consider the function u(x) to be defined for x > 0.
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