Combination Theorem for Continuous Mappings/Metric Space

Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $\R$ denote the real numbers.

Let $f: M \to \R$ and $g: M \to \R$ be real-valued functions from $M$ to $\R$ which are continuous on $M$.

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

Then the following results hold:

Sum Rule

$f + g$ is ‎continuous on $M$.

Difference Rule

$f - g$ is ‎continuous on $M$.

Multiple Rule

$\lambda f$ is ‎continuous on $M$.

Combined Sum Rule

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

Product Rule

$f g$ is ‎continuous on $M$.

Quotient Rule

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

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

Absolute Value Rule

$\size f$ is continuous at $a$

where:

$\map {\size f} x$ is defined as $\size {\map f x}$.

Maximum Rule

$\max \set {f, g}$ is ‎continuous on $M$.

Minimum Rule

$\min \set {f, g}$ is ‎continuous on $M$.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 14$