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 $$\mathbb C$$) 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
 * $$f: \mathbb C \to \mathbb C, f(z) := z^2$$

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
 * $$\lim_{z \to z_0 \atop z \in A} f(z) = w$$

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

Proof: Let $$\epsilon > 0$$ be arbitrary. Since
 * $$\lim_{z \to z_0 \atop z \in A} f(z) = w$$,

there exists a $$\delta > 0$$ such that
 * $$z \in A \cap B(z_0, \delta) \Rightarrow |f(z) - w| < \epsilon$$.

But since $$B \subseteq A$$, we also have $$B \cap B(z_0, \delta) \subseteq A \cap B(z_0, \delta)$$, and thus
 * $$z \in B \cap B(z_0, \delta) \Rightarrow z \in A \cap B(z_0, \delta) \Rightarrow |f(z) - w| < \epsilon$$,

and therefore
 * $$\lim_{z \to z_0 \atop z \in B} f(z) = w$$.

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 $$\epsilon > 0$$ be arbitrary. As
 * $$\lim_{z \to z_0 \atop z \in O} f(z) = w$$,

there exists a $$\delta_2 > 0$$ such that
 * $$z \in B(z_0, \delta_2) \cap U \Rightarrow |f(z) - f(z_0)| < \epsilon$$.

We define $$\delta := \min\{\delta_1, \delta_2\}$$ and obtain
 * $$z \in B(z_0, \delta) \cap A \Rightarrow z \in B(z_0, \delta) \Rightarrow z \in B(z_0, \delta_2) \cap U \Rightarrow |f(z) - f(z_0)| < \epsilon$$.

Exercises
 Prove that if we define
 * $$f: \mathbb C \to \mathbb C, f(z) = \begin{cases}

\frac{z^2}{|z|^2} & z \neq 0 \\ 1 & z = 0 \end{cases}$$, then $$f$$ is not continuous at $$0$$. Hint: Consider the limit with respect to different lines through $$0$$ and use theorem 2.2.4.  Next