Combination Theorem for Continuous Mappings/Topological Ring/Sum Rule

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

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

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

Let $f + g : S \to R$ be the mapping defined by:
 * $\forall x \in S: \map {\paren{f + g}} x = \map f x + \map g x$

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

Proof
By definition of a topological ring, $\struct {R, +, \tau_{_R} }$ is a topological group.

From Product Rule for Continuous Mappings to Topological Group:
 * $f + g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.

Also see

 * Product Rule for Continuous Mappings to Topological Group