# Real function/Continuous/Rules/Section

## Lemma

Let ${\displaystyle {}D\subseteq \mathbb {R} }$ and ${\displaystyle {}E\subseteq \mathbb {R} }$ be subsets and let

${\displaystyle f\colon D\longrightarrow \mathbb {R} }$

and

${\displaystyle g\colon E\longrightarrow \mathbb {R} }$

denote functions with

${\displaystyle {}f(D)\subseteq E}$. Then the following statements hold.
1. If ${\displaystyle {}f}$ is continuous in ${\displaystyle {}x\in D}$ and ${\displaystyle {}g}$ is continuous in ${\displaystyle {}f(x)}$, then also the composition ${\displaystyle {}g\circ f}$ is continuous in ${\displaystyle {}x}$.
2. If ${\displaystyle {}f}$ and ${\displaystyle {}g}$ are continuous, so is ${\displaystyle {}g\circ f}$.

### Proof

The first statement follows from fact. This implies also the second statement.

${\displaystyle \Box }$

## Lemma

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

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

be continuous functions. Then also the functions

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

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

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

is continuous.

### Proof

This follows from fact and fact.

${\displaystyle \Box }$

## Corollary

${\displaystyle P\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto P(x),}$
are

### Proof

Due to example and fact, the powers

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},}$

are continuous for every ${\displaystyle {}n\in \mathbb {N} }$. Hence, also the functions

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax^{n},}$

are continuous for every ${\displaystyle {}a\in \mathbb {R} }$ and therefore, again due to fact, the functions

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0},}$

are continuous.

${\displaystyle \Box }$

## Corollary

Let ${\displaystyle {}P,Q\in \mathbb {R} [X]}$ be polynomials and let ${\displaystyle {}U:={\left\{x\in \mathbb {R} \mid Q(x)\neq 0\right\}}}$. Then the rational function

${\displaystyle U\longrightarrow \mathbb {R} ,x\longmapsto {\frac {P(x)}{Q(x)}},}$

is continuous.

### Proof

This follows from fact and fact.

${\displaystyle \Box }$