Combination Theorem for Continuous Mappings/Metric Space/Multiple Rule

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Theorem

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

Let $\R$ denote the real numbers.

Let $f: M \to \R$ be a real-valued function from $M$ to $\R$ which is continuous on $M$.

Let $\lambda \in \R$ be an arbitrary real number.

Then:

$\lambda f$ is ‎continuous on $M$.

Proof

By definition of ‎continuous:

$\forall a \in M: \ds \lim_{x \mathop \to a} \map f x = \map f a$

Let $f$ tend to the following limit:

$\ds \lim_{x \mathop \to a} \map f x = l$

From the Multiple Rule for Limits of Real Functions, we have that:

$\ds \lim_{x \mathop \to a} \paren {\lambda \map f x} = \lambda l$

So, by definition of ‎continuous again, we have that $\lambda \map f x$ is continuous on $M$.

$\blacksquare$