Combination Theorem for Continuous Functions/Real

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Let $\R$ denote the real numbers.

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

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

Then the following results hold:

Sum Rule

$f + g$ is ‎continuous on $S$.

Difference 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