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__Mathematics of Classical and Quantum Mechanics__by Byron and Fuller, they state that "Some authors (never mathematicians) define an analytic function as a differentiable function

*with a continuous derivative*." ..."But this is a mathematical fraud of cosmic proportions.. "

Their main point is that you don't have to assume continuity of the first derivatives of an analytic function to prove Cauchy's Integral Theorem if you use the Goursat approach, yet I thought that really IS how an analytic function is defined, i.e. that a function of a complex variable is analytic within a region S if it is differentiable within and on the boundary of S.