Combination Theorem for Continuous Functions/Complex/Multiple 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 $S \subseteq \C$.
Let $\lambda \in \C$ be an arbitrary complex number.
Then:
- $\lambda f$ is continuous on $S$.
Proof
By definition of continuous, we have that
- $\forall c \in S: \ds \lim_{z \mathop \to c} \map f z = \map f c$
Let $f$ tend to the following limit:
- $\ds \lim_{z \mathop \to c} \map f z = l$
From the Multiple Rule for Limits of Complex Functions, we have that:
- $\ds \lim_{z \mathop \to c} \paren {\lambda \map f z} = \lambda l$
So, by definition of continuous again, we have that $\lambda f$ is continuous on $S$.
$\blacksquare$