Combination Theorem for Continuous Functions/Real/Multiple Rule
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Theorem
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions which are continuous on an open subset $S \subseteq \R$.
Let $\lambda \in \R$ be an arbitrary real number.
Then:
- $\lambda f$ is continuous on $S$.
Proof
By definition of continuous, we have that
- $\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
Let $f$ tend to the following limit:
- $\ds \lim_{x \mathop \to c} \map f x = l$
From the Multiple Rule for Limits of Real Functions, we have that:
- $\ds \lim_{x \mathop \to c} \paren {\lambda \map f x} = \lambda l$
So, by definition of continuous again, we have that $\lambda f$ is continuous on $S$.
$\blacksquare$