Combination Theorem for Continuous Functions/Real/Combined Sum Rule

Theorem
Let $X$ be one of the standard number fields $\Q, \R, \C$.

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

Let $\lambda, \mu \in X$ be arbitrary numbers in $X$.

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

Proof
By definition of continuous, we have that
 * $\forall c \in S: \displaystyle \lim_{x \mathop \to c} f \left({x}\right) = f \left({c}\right)$
 * $\forall c \in S: \displaystyle \lim_{x \mathop \to c} g \left({x}\right) = g \left({c}\right)$

Let $f$ and $g$ tend to the following limits:
 * $\displaystyle \lim_{x \mathop \to c} f \left({x}\right) = l$
 * $\displaystyle \lim_{x \mathop \to c} g \left({x}\right) = m$

From the Combined Sum Rule for Limits of Functions, we have that:
 * $\displaystyle \lim_{x \mathop \to c} \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda l + \mu m$

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