Real function/Continuous/Addition, multiplication, inverting/Fact

Rules for continuous functions

Let ${}D\subseteq \mathbb {R}$ be a subset and let

$f,g\colon D\longrightarrow \mathbb {R}$

be continuous functions.

Then also the functions

$f+g\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)+g(x),$

$f-g\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)-g(x),$

$f\cdot g\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)\cdot g(x),$

are continuous. For a subset ${}U\subseteq D$ such that ${}g$ has no zero in ${}U$, also

$f/g\colon U\longrightarrow \mathbb {R} ,x\longmapsto f(x)/g(x),$

is continuous.

Proof, Write another proof