Combination Theorem for Continuous Functions/Real/Combined Sum Rule

Theorem
Let $\R$ denote the real numbers.

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

Let $\lambda, \mu \in \R$ be arbitrary real numbers.

Then:
 * $\lambda f + \mu g$ is continuous on $S$.

Proof
By definition of continuous, we have that
 * $\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
 * $\forall c \in S: \ds \lim_{x \mathop \to c} \map g x = \map g c$

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 Combined Sum Rule for Limits of Real Functions, we have that:
 * $\ds \lim_{x \mathop \to c} \paren {\lambda \map f x + \mu \map g x} = \lambda l + \mu m$

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