# Combination Theorem for Continuous Mappings/Metric Space

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## 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 $2.6: 14$