Combination Theorem for Continuous Functions/Complex/Sum 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 $S$.


Proof

By definition of ‎continuous:

$\forall c \in S: \ds \lim_{x \mathop \to c} \map f z = \map f c$
$\forall c \in S: \ds \lim_{x \mathop \to c} \map g z = \map g c$


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 Sum Rule for Limits of Complex Functions, we have that:

$\ds \lim_{z \mathop \to c} \paren {\map f z + \map g z} = l + m$


So, by definition of ‎continuous again, we have that $f + g$ is continuous on $S$.

$\blacksquare$