Complex Variables : Differentiability
November 16th, 2022
9. Let f denote the function whose values are
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.)