Continuity of Complex-Valued Functions, and the Modulus of a Power

In the setting of Complex Analysis: Standing Notation, let $(X,d)$ be a metric space, let $A\subseteq X$, let $a\in A$, let $f,g:A\to\mathbb{C}$ be functions, and let $\lambda\in\mathbb{C}$. Continuity at $a$ relative to $A$ is as in Continuous Map Between Metric Spaces (i), for $d$ on $X$, $d_{\mathbb{C}}$ on $\mathbb{C}$ and the distance $d_{\mathbb{R}}$ of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv) on $\mathbb{R}$. Sums, scalar multiples, products and quotients of functions $A\to\mathbb{C}$, and the functions $\operatorname{Re}f,\operatorname{Im}f,|f|:A\to\mathbb{R}$ and $\overline{f}:A\to\mathbb{C}$, are defined pointwise.

(i) The modulus of a power. Let $z\in\mathbb{C}$ and let $n\in\mathbb{N}_{0}$; this clause involves none of $(X,d)$, $A$, $a$, $f$, $g$ and $\lambda$. Then

$$|z^{n}|=|z|^{n}.$$

(ii) Constants and the identity. For every $c\in\mathbb{C}$, the constant function $A\to\mathbb{C}$ with value $c$ is continuous at $a$ relative to $A$. If $(X,d)=(\mathbb{C},d_{\mathbb{C}})$, the function $A\to\mathbb{C}$, $w\mapsto w$, is continuous at $a$ relative to $A$.

(iii) Sums, products and scalar multiples. If $f$ and $g$ are continuous at $a$ relative to $A$, then so are $f+g$, $fg$ and $\lambda f$.

(iv) Reciprocals. If $g(x)\neq0$ for every $x\in A$ and $g$ is continuous at $a$ relative to $A$, then $1/g$ is continuous at $a$ relative to $A$.

(v) Composition. Let $B\subseteq\mathbb{C}$ with $f(x)\in B$ for every $x\in A$, and let $h:B\to\mathbb{C}$. If $f$ is continuous at $a$ relative to $A$ and $h$ is continuous at $f(a)$ relative to $B$, then $h\circ f:A\to\mathbb{C}$ is continuous at $a$ relative to $A$.

(vi) Parts, modulus and conjugate. If $f$ is continuous at $a$ relative to $A$, then so are $\operatorname{Re}f$, $\operatorname{Im}f$, $|f|$ and $\overline{f}$. Conversely, $f$ is continuous at $a$ relative to $A$ if $\operatorname{Re}f$ and $\operatorname{Im}f$ are.