Complex Analysis/Limits and continuity of complex functions

In this section, we

• introduce a 'broader class of limits' than known from real analysis (namely limits with respect to a subset of $ℂ$) and
• characterise continuity of functions mapping from a subset of the complex numbers to the complex numbers using this 'class of limits'.

Complex functions

Example 2.2:

The function

$f:ℂ\to ℂ,f(z):=z^2$

is a complex function.

Limits of complex functions with respect to subsets of the preimage

We shall now define and deal with statements of the form

$\underset{\genfrac{}{}{0ex}{}{z\to z_0}{z\in A}}{\lim }f(z)=w$

for $S\subseteq ℂ$, $f:S\to ℂ$, $A\subseteq S$ and $w\in ℂ$, and prove two lemmas about these statements.

Proof: Let $ϵ>0$ be arbitrary. Since

$\underset{\genfrac{}{}{0ex}{}{z\to z_0}{z\in A}}{\lim }f(z)=w$,

there exists a $\delta >0$ such that

$z\in A\cap B(z_0,\delta )\Rightarrow |f(z)-w|<ϵ$.

But since $B\subseteq A$, we also have $B\cap B(z_0,\delta )\subseteq A\cap B(z_0,\delta )$, and thus

$z\in B\cap B(z_0,\delta )\Rightarrow z\in A\cap B(z_0,\delta )\Rightarrow |f(z)-w|<ϵ$,

and therefore

$\underset{\genfrac{}{}{0ex}{}{z\to z_0}{z\in B}}{\lim }f(z)=w$.$◻$

Proof:

Let $A\subseteq S$ such that $z_0\in A$.

First, since $O$ is open, we may choose ${\delta }_1>0$ such that $B(z_0,{\delta }_1)\subseteq O$.

Let now $ϵ>0$ be arbitrary. As

$\underset{\genfrac{}{}{0ex}{}{z\to z_0}{z\in O}}{\lim }f(z)=w$,

there exists a ${\delta }_2>0$ such that

$z\in B(z_0,{\delta }_2)\cap U\Rightarrow |f(z)-f(z_0)|<ϵ$.

We define $\delta :=\min \{{\delta }_1,{\delta }_2\}$ and obtain

$z\in B(z_0,\delta )\cap A\Rightarrow z\in B(z_0,\delta )\Rightarrow z\in B(z_0,{\delta }_2)\cap U\Rightarrow |f(z)-f(z_0)|<ϵ$.$◻$

Exercises

1. Prove that if we define

   $f:ℂ\to ℂ,f(z)=\{\begin{array}{cc}\frac{z^2}{|z{|}^2}& z\ne 0\\ 1& z=0\end{array}$,

   then $f$ is not continuous at $0$. Hint: Consider the limit with respect to different lines through $0$ and use theorem 2.2.4.

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