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My complex analysis textbook emphasizes that the Cauchy-Goursat theorem is true even if the function we are integrating does not have continuous first partials and that this differs from the real case.

Looking over the proof in my textbook, I don't really see anything being done that couldn't be done in R^2. So why do we need the first partials to be continuous when we integrate over a closed curve in R^2 but not when we integrate over a closed contour in C?