f(z) =|z| Show that the function f(z) = x+4iy is not differentiable at any point z. Solution Let z be any point in the complex plane. With Az = Ax+iAy, f(z+Az)f(z) = (x + Ax)+4i(y+ Ay)-x-4iy = Ax+4iAy and so f(z+ Az) − f(z) lim Az-0 Δε Ax+4iAy = lim Az-0 Ax+iAy (9) Now, as shown in FIGURE 3.2.1(a), if we let Az → 0 along a line parallel to the x-axis, then Ay = 0 and Az = Ax. Along this path we have: f(z+Az) − f(z) Ax lim Az-0 Δε lim Az-0 Ax = 1. (10) On the other hand, if we let Az → 0 along a line parallel to the y-axis as shown in Figure 3.2.1(b), then Ax = 0 and Az = i Ay so that lim A:-0 f(z+ Az) − f(z) Δε 4i Ay = lim A-0 ¡Ay = 4. (11) In view of the obvious fact that the values in (10) and (11) are different, we conclude that the limit in (9) does not exist. Therefore, f(z) = x+4iy is nowhere differentiable; that is, f is not differentiable at any point z. Az = Ax (a) Az 0 along a line parallel to x-axis Az - Ay (b) A 0 along a line parallel to y-axis