Toronto Math Forum

MAT334--2020F => MAT334--Tests and Quizzes => Quiz 2 => Topic started by: Xinqiao Li on October 03, 2020, 05:47:44 AM

Title: Quiz2 LEC0101 C
Post by: Xinqiao Li on October 03, 2020, 05:47:44 AM
Questions: Find all points of continuity of the given function: $f(z)=(Imz-Rez)^{-1}$

Solutions:

Let $z=x+iy$ where x,y are real numbers.

Then $f(z)=f(x,y)=(Im(x+iy)-Re(x+iy))^{-1}=(y-x)^{-1}=\frac{1}{y-x}$

The function is not valid when the denominator equals 0, that is, $y=x$.

Therefore, the function is discontinuous only at all points of the line $y=x$.