Combination Theorem for Continuous Functions/Complex/Multiple Rule

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 $S$.

$\blacksquare$