Combination Theorem for Continuous Functions

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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 $\map g x \ne 0$.


Also see


Sources