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 $M$.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 14$