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I am attempting to self-study Complex Analysis but i am confused over a couple of points.

1 - my book says "if a complex function is differentiable once throughout its domain of definition then it is infinitely differentiable" . How does this apply to z^2 ? If you differentiate it once you get 2z, differentiating again gives 2 and once more gives 0.

2- My book states that this infinitely differentiable point contrasts with the real case and gives an example of f(x)=0 for x≤0 and f(x) = x^2 for x≥0 and then goes on to say the second derivative at 0 does not exist, being 0 from the left and 2 from the right. But it seems to me that the same would happen for z^2 ?

Thanks

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# Infinitely differentiable complex functions

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