Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 15

Warm-up-exercises

Exercise

Show that a linear function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax,$

is continuous.

Exercise

Prove that the function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \vert {x}\vert ,$

is continuous.

Exercise

Prove that the function

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},x\longmapsto {\sqrt {x}},$

is continuous.

Exercise

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

$f\colon T\longrightarrow \mathbb {R}$

be a continuous function. Let ${}x\in T$ be a point such that ${}f(x)>0$. Prove that ${}f(y)>0$ for all ${}y$ in a non-empty open interval ${}]x-a,x+a[$.

Exercise

Let ${}a<b<c$ be real numbers and let

$f\colon [a,b]\longrightarrow \mathbb {R}$

and

$g\colon [b,c]\longrightarrow \mathbb {R}$

be continuous functions such that ${}f(b)=g(b)$. Prove that the function

$h\colon [a,c]\longrightarrow \mathbb {R}$

such that

$h(t)=f(t){\text{ for }}t\leq b{\text{ and }}h(t)=g(t){\text{ for }}t>b$

is also continuous.

Exercise

Compute the limit of the sequence

${}x_{n}=5\left({\frac {2n+1}{n}}\right)^{3}-4\left({\frac {2n+1}{n}}\right)^{2}+2\left({\frac {2n+1}{n}}\right)-3\,$

for ${}n\rightarrow \infty$.

Exercise

Let

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

be a continuous function which takes only finitely many values. Prove that ${}f$ is constant.

Exercise

Give an example of a continuous function

$f\colon \mathbb {Q} \longrightarrow \mathbb {R} ,$

which takes exactly two values​​.

Exercise

Prove that the function

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

defined by

${}f(x)={\begin{cases}x,\,{\text{ if }}x\in \mathbb {Q} \,,\\0,\,{\text{ otherwise}}\,,\end{cases}}\,$

is only at the zero point ${}0$ continuous.

Exercise

Let ${}T\subseteq \mathbb {R}$ be a subset and let ${}a\in \mathbb {R}$ be a point. Let ${}f\colon T\rightarrow \mathbb {R}$ be a function and ${}b\in \mathbb {R}$. Prove that the following statements are equivalent.

1. We have

   ${}\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.$

2. For all ${}\epsilon >0$ there exists a ${}\delta >0$ such that for all ${}x\in T$ with ${}d(x,a)\leq \delta$ the inequality ${}d(f(x),b)\leq \epsilon$ holds.

Hand-in-exercises

Exercise (2 marks)

We consider the function

${}f(x)={\begin{cases}1{\text{ for }}x\leq -1\,,\\x^{2}{\text{ for }}-1<x<2\,,\\-2x+7{\text{ for }}x\geq 2\,.\end{cases}}\,$

Determine the points ${}x\in \mathbb {R}$ where ${}f$ is continuous.

Exercise (3 marks)

Compute the limit of the sequence

${}b_{n}=2a_{n}^{4}-6a_{n}^{3}+a_{n}^{2}-5a_{n}+3\,,$

where

${}a_{n}={\frac {3n^{3}-5n^{2}+7}{4n^{3}+2n-1}}\,.$

Exercise (3 marks)

Prove that the function ${}f\colon \mathbb {R} \rightarrow \mathbb {R}$ defined by

${}f(x)={\begin{cases}1,{\text{ if }}x\in \mathbb {Q} \,,\\0\,{\text{ otherwise}}\,,\end{cases}}\,$

is for no point ${}x\in \mathbb {R}$ continuous.

Exercise (3 marks)

Decide whether the sequence

${}a_{n}={\sqrt {n+1}}-{\sqrt {n}}\,$

converges and in case determine the limit.

Exercise (4 marks)

Determine the limit of the rational function

${\frac {2x^{3}+3x^{2}-1}{x^{3}-x^{2}+x+3}}$

in the point ${}a=-1$.