Question from Complex Analysis Show that Log z is not contin

Question from Complex Analysis

Show that Log z is not continuous on the negative real axis including the point x=0

Solution

The function Log z is not continuous on the negative real axis {z = x + iy : x < 0, y = 0} (Unlike real logarithm, it is defined there, but useless).

To see this consider the point z = , > 0. Consider the sequences {a_n = e^{i( 1 n )}} and {b_n = e^{i(+ 1 n )}}. Then lim_n a_n = lim_n b_n = z but lim_n Log a_n = lim_n ln +i( 1/n ) = ln +i and

lim_n Log b_n = ln i. That is limit n for log z is not equal on both.