MAT334--2020F > Quiz 2

Quiz2-6101 6D

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Junhong Zhou:
Problem(3pt). Find all points of continuity of the given function;
$$f(z)= \begin{cases}\frac{z^4-1}{z-i},& z\neq i\\4i, & z=i \end{cases}$$

f(z) is continuous when $z\neq i$.

When z = i, then

$z^4-1=i^4-1=1-1=0$

$z - i = i-i = 0$

Now use the L'Hospital's Rule we have:
\begin{align*}
\lim_{z \to i} \frac{z^4-1}{z-i} &= \lim_{z \to i} \frac{4z^3}{1}\\
&= 4i^3\\
&= -4i\\
& \neq 4i
\end{align*}
Therefore f(z) is not continuous at z = i.