Combination Theorem for Continuous Functions

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 the following results hold:

Sum Rule

$f + g$ is continuous on $S$.

Multiple Rule

$\lambda f$ is continuous on $S$.

Combined Sum Rule

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

Product Rule

$f g$ is continuous on $S$

Quotient Rule

$\dfrac f g$ is continuous on $S \setminus \set {x \in S: \map g x = 0}$

that is, on all the points $x$ of $S$ where $\map g x \ne 0$.