Lecture 12

\(*\) \(\Omega \subseteq \C\) is
*piecewise affinely path connected* if any two points in \(\Omega\) can be connected by a
finite number of line segments *in \(\Omega\)*, joined end to end.

For open sets in \(\C\), the two definitions are equivalent

(not so in general: see for example the comb space, to be confirmed/cleaned-up).

**Show:** If \(\Omega_1,\Omega_2 \subseteq \C\) are open, then so is \(\Omega_1\cap
\Omega_2\).

**Proof:** If \(\Omega_1\cap \Omega_2 =
\emptyset\) done
(\(\emptyset\) is open).

Otherwise, for any \(z\in \Omega_1\cap\Omega_2\) we have

\(z\in \Omega_1 \Rightarrow \exists\varepsilon_1 \gt0 \text{ such that } B_{\varepsilon_1}(z)\subset \Omega_1\) ... ♣️

\(z\in \Omega_1 \Rightarrow \exists\varepsilon_2 \gt 0 \text{ such that } B_{\varepsilon_2}(z)\subset \Omega_2\) ... 🙂

Since \(\Omega_1,\Omega_2\) are open, set \(\varepsilon = \min \{\varepsilon_1, \varepsilon_2\}\) and note that \(\varepsilon>0\).

Thus

\(B_{\varepsilon}(z)\subset \Omega_1\) by ♣️,

\(B_{\varepsilon}(z)\subset \Omega_2\) by 🙂.

So \(B_{\varepsilon}(z)\subset \Omega_1\cap\Omega_2\). Since \(z\) was arbitrary in \(\Omega_1\cap\Omega_2\).

This implies \(\text{Int } \Omega_1\cap\Omega_2= \Omega_1\cap\Omega_2\). Hence \(\Omega_1\cap\Omega_2\) is open. \(\blacksquare\)

\(*\) An open, connected subset of \(\C\) is called *domain*.

\(*\) An set whose interior is a domain is called a *region*.

\(*\) A Point \(z\in \C\) is called an
*accumulation point* of a set \(\Omega \subseteq \C\) if every deleted neighbouhood
of \(z\) intersects \(\Omega\).

**Example 1:**
\(\Omega=\left\{\dfrac{i}{2^n}\right\}_{n\in \N}\).
The only accumulation point is 0.

**Example 2:**
\(\Omega=B_1\).
The set of accumulation points is \(\conj{B_1}\).

Let \(f\) be a \(\C\)-valued function defined on a deleted neighbourhood of \(z_0\in\C\).

\(\ds \lim_{z\ra z_0}f(z)=w_0\) says: Given \(\varepsilon > 0\) exists \(\delta > 0\) such that \[0<|z-z_0|<\delta \implies \left|f(z)-w_0\right|<\varepsilon.\]

**Note:** \(f\) does not have to be defined
at \(z_0\).

Examples:

\( f(z) = \left\{ \begin{array}{ll} 0 & z\neq 0,\\ 1337 & z=0. \end{array} \right. \) \(\ds\lim_{z\ra 0}f(z)=0\);

\[\ds \lim_{z\ra 0}\dfrac{\sin z}{z}=1.\]

**Remark:** If a
limit exists, it is unique.

Suppose \(f(z) = f(x+iy) = u(x,y) + i v(x,y) \), \[z_0= x_0+iy_0 \text{ and } w_0 =u_0+iv_0.\]

**Theorem 1:**
\[
\lim_{z \ra z_0} f(z)=w_0 \iff
\left\{
\begin{array}{rl}
\ds\lim_{(x,y)\ra (x_0, y_0)} u(x,y) & = u_0,\\
\ds\lim_{(x,y)\ra (x_0, y_0)} v(x,y) & = v_0.
\end{array}
\right.
\]

**Theorem 2:** Suppose \(\ds \lim_{z \ra z_0}f(z) = w_0\)
and \(\ds \lim_{z \ra z_0}g(z) = \xi_0\) and \(\lambda\in \C\). Then

- \(\ds \lim_{z \ra z_0}\left(f \pm g\right)(z) = w_0 \pm \xi_0\);
- \(\ds \lim_{z \ra z_0}\left(\lambda \cdot f \right)(z) = \lambda \cdot w_0\);
- \(\ds \lim_{z \ra z_0}\left(f \cdot g\right)(z) = w_0 \cdot \xi_0\);
- \(\ds \lim_{z \ra z_0}\dfrac{f(z)}{g(z)} = \dfrac{w_0}{\xi_0}\), as long as \(\xi_0\neq 0\).