Complex Variables : Differentiability

9. Let f denote the function whose values are

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f (z) = z^2 / z when z ≠ 0,

f (z) = 0 when z = 0.

Show that if z = 0, then ∆w/∆z = 1 at each nonzero point on the real and imaginary axes in the ∆z, or ∆x∆y, plane. Then show that ∆w/∆z = -1 at each nonzero point (∆x, ∆x) on the line ∆y = ∆x in that plane. Conclude from these observations that f ΄(0) does not exist.

(Note that, to obtain this result, it is not sufficient to consider only horizontal and vertical approaches to the origin in the ∆z plane.)