MTH 305: COMPLEX ANALYSIS I. Answer FOUR questions only.

Question

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To establish the continuity of the function $f(z)$ at $z=i$ , we need to check three conditions:

The given function is:

f(z) = \begin{cases} z^2, & \text{if } z \neq i \\ 0, & \text{if } z = i \end{cases}

Evaluate $f(i)$ .
According to the definition of the function, when $z=i$ , $f(z)=0$ .
$f(i) = 0$
The function is defined at $z=i$ .
Evaluate $\lim_{z \to i} f(z)$ .
As $z$ approaches $i$ , $z$ is not equal to $i$ . Therefore, we use the first part of the function definition, $f(z) = z^2$ .
$\lim_{z \to i} f(z) = \lim_{z \to i} z^2$
Substitute $z=i$ into the expression:
$\lim_{z \to i} z^2 = i^2 = -1$
The limit exists and is equal to $-1$ .
Compare $f(i)$ with $\lim_{z \to i} f(z)$ .
We have $f(i) = 0$ and $\lim_{z \to i} f(z) = -1$ .
Since $f(i) \neq \lim_{z \to i} f(z)$ , the third condition for continuity is not met.
Conclude on continuity.
The function $f(z)$ is not continuous at $z=i$ . For all other points $z \neq i$ , $f(z) = z^2$ , which is a polynomial and thus continuous everywhere. Therefore, the function is continuous for all $z \neq i$ but discontinuous at $z=i$ .

The function is discontinuous at $z=i$ .

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