# Combination Theorem for Continuous Mappings/Topological Ring

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## Theorem

Let $\struct{S, \tau_{_S}}$ be a topological space.

Let $\struct{R, +, *, \tau_{_R}}$ be a topological ring.

Let $\lambda \in R$.

Let $f,g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings.

Then the following results hold:

### Sum Rule

- $f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

### Translation Rule

- $\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

### Negation Rule

- $-g : \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

### Product Rule

- $f * g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.

### Multiple Rule

- $\lambda * f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous
- $f * \lambda: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.