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$ Let f(z) = (Re(z))^3 + (Im(z))^3$

(a) At which points (if any) is f differentiable? Find the expression of f' at those points.

(b) Draw a picture of the subset of C consisting of those points at which f is differentiable. Hence decide at which points (if any) f is analytic.

Answer:

$ f(z) = x^3 + y^3 $

(a) Using Cauchy Riemann equations I found that f is only differentiable at (0,0). And f' is 0 at that point.

(b) f is nowhere analytic as it is not differentiable in an $\epsilon$ neighbourhood of z.

Is that correct?

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