Combination Theorem for Continuous Functions/Multiple Rule

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Theorem

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 $X$.


Then:

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


Proof

By definition of continuous, we have that

$\forall c \in S: \displaystyle \lim_{x \mathop \to c} \ f \left({x}\right) = f \left({c}\right)$


Let $f$ tend to the following limit:

$\displaystyle \lim_{x \mathop \to c} \ f \left({x}\right) = l$


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

$\displaystyle \lim_{x \mathop \to c} \ \left({\lambda f \left({x}\right)}\right) = \lambda l$


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

$\blacksquare$