Combination Theorem for Continuous Mappings/Metric Space/Combined Sum Rule

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:

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

Proof

From the Multiple Rule for Continuous Mappings on Metric Space, we have that:

$\lambda \map f x$ and $\mu \map g x$ are ‎continuous.

From the Sum Rule for Continuous Mappings on Metric Space, we have that:

$\lambda \map f x + \mu \map g x$ is ‎continuous.

So, by definition of ‎continuous again, we have that $\lambda f + \mu g$ is continuous on $M$.

$\blacksquare$