Combination Theorem for Continuous Functions/Real/Sum Rule

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Let $\R$ denote the real numbers.

Let $f$ and $g$ be real functions which are continuous on an open subset $S \subseteq \R$.


$f + g$ is ‎continuous on $S$.


By definition of ‎continuous:

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

Let $f$ and $g$ tend to the following limits:

$\ds \lim_{x \mathop \to c} \map f x = l$
$\ds \lim_{x \mathop \to c} \map g x = m$

From the Sum Rule for Limits of Real Functions, we have that:

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

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