Combination Theorem for Continuous Functions/Complex/Product Rule
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Theorem
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions which are continuous on an open subset $S \subseteq \C$.
Then:
- $f g$ is continuous on $S$
where $f g$ denotes the pointwise product of $f$ and $g$.
Proof
By definition of continuous:
- $\forall c \in S: \ds \lim_{z \mathop \to c} \map f z = \map f c$
- $\forall c \in S: \ds \lim_{z \mathop \to c} \map g z = \map g c$
Let $f$ and $g$ tend to the following limits:
- $\ds \lim_{z \mathop \to c} \map f z = l$
- $\ds \lim_{z \mathop \to c} \map g z = m$
From the Product Rule for Limits of Complex Functions:
- $\ds \lim_{z \mathop \to c} \paren {\map f z \map g z} = l m$
So, by definition of continuous again, we have that $f g$ is continuous on $S$.
$\blacksquare$