Combination Theorem for Continuous Functions/Multiple Rule

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Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $f$ be a function which is continuous on an open subset $S \subseteq X$.

Let $\lambda \in X$ be an arbitrary number in $X$.


$\lambda f$ is continuous on $S$.


By definition of continuous, we have that

$\forall c \in S: \displaystyle \lim_{x \mathop \to c} \map f x = \map f c$

Let $f$ tend to the following limit:

$\displaystyle \lim_{x \mathop \to c} \map f x = l$

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

$\displaystyle \lim_{x \mathop \to c} \paren {\lambda \, \map f x} = \lambda l$

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