# 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.
* We have
    $\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.$
* 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 $a=-1$.