Toronto Math Forum
MAT3342020F => MAT334Tests and Quizzes => Quiz 2 => Topic started by: Junhong Zhou on October 02, 2020, 02:22:25 PM

Problem(3pt). Find all points of continuity of the given function;
$$f(z)=
\begin{cases}\frac{z^41}{zi},& z\neq i\\4i, & z=i
\end{cases}$$
Answer:
f(z) is continuous when $z\neq i$.
When z = i, then
$z^41=i^41=11=0$
$z  i = ii = 0$
Now use the L'Hospital's Rule we have:
\begin{align*}
\lim_{z \to i} \frac{z^41}{zi} &= \lim_{z \to i} \frac{4z^3}{1}\\
&= 4i^3\\
&= 4i\\
& \neq 4i
\end{align*}
Therefore f(z) is not continuous at z = i.