Show that the function f(z) = xy + iy is continuous everywhere but not

1.	Show that the function f(z) = xy + iy is continuous everywhere but not differentiable anywhere.
Answer» Given f(z) = xy + iy u = xy, v = y x and y are continuous functions ,therefore u and v are also continuous. But u = xy, v = y u_x = y, v_x = 0 u_y = x, v_y = 1 u_x ≠ v_y and v_x - u_y C-R equations are not satisfied. Hence f(z) is not differentiable anywhere though it is continuous everywhere.

Show that the function f(z) = xy + iy is continuous everywhere but not differentiable anywhere.

Answer»

Given f(z) = xy + iy

u = xy, v = y

x and y are continuous functions ,therefore u and v are also continuous.

But u = xy, v = y

u_x = y, v_x = 0

u_y = x, v_y = 1

u_x ≠ v_y and v_x - u_y

C-R equations are not satisfied.

Hence f(z) is not differentiable anywhere though it is continuous everywhere.

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Show that the function f(z) = xy + iy is continuous everywhere but not differentiable anywhere.

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