# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 10

    Exercises

### Exercise

Show that a linear function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax,$
is continuous.

### Exercise

Let $D\subseteq \mathbb {R}$ be a subset, $f\colon D\rightarrow \mathbb {R}$ a function and $a\in D$ a point. Show that the following properties are equivalent.
* $f$ is continuous in $a$.
* For every $n\in \mathbb {N} _{+}$, there exists some $m\in \mathbb {N} _{+}$ such that for the estimate holds.
    $\vert {x-a}\vert \leq {\frac {1}{m}}\,$
    $\vert {f(x)-f(a)}\vert \leq {\frac {1}{n}}\,$
* For every $s\in \mathbb {N}$, there exists some $r\in \mathbb {N}$ such that for the estimate holds.
    $\vert {x-a}\vert \leq {\frac {1}{10^{r}}}\,$
    $\vert {f(x)-f(a)}\vert \leq {\frac {1}{10^{s}}}\,$

### 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

Farmer Ernst wants to make a square-shaped field for melons. The field shall be $100$ square meters in size, but he thinks that a size between $99$ and $101$ square meters is acceptable. What error is allowed for the side lengths in order to stick within this tolerance?

### Exercise

Let
    $f(x)=2x^{3}-4x+5\,.$
Show that for all $x\in \mathbb {R}$, the following relation holds: If
    $\vert {x-3}\vert \leq {\frac {1}{800}}\,,$
then
    $\vert {f(x)-f(3)}\vert \leq {\frac {1}{10}}\,.$

### Exercise

For the function
    $f(x)=2x^{3}-4x^{2}+x-6\,$
and the point $a=1$, determine for $\epsilon ={\frac {1}{10}}$ an explicit $\delta >0$ such that
    $d(x,a)\leq \delta \,$
implies the estimate
    $d(f(x),f(a))\leq \epsilon \,.$

### 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 $a\in \mathbb {R}$ and let
    $f,g\colon \mathbb {R} \longrightarrow \mathbb {R}$
be continuous functions with
    $f(a)>g(a)\,.$
Show that there exists some $\delta >0$ such that
    $f(x)>g(x)\,$
holds for all $x\in [a-\delta ,a+\delta ]$.

### Exercise

Let $f\colon \mathbb {R} \rightarrow \mathbb {R}$ be a continuous function. Show the following statements.
* The function $f$ is uniquely determined by its values on $\mathbb {Q}$.
* The value $f(a)$ is determined by the values $f(x)$, $x\neq a$.
* If for all $x<a$, the estimate holds, then also holds.
    $f(x)\leq c\,$
    $f(a)\leq c\,$

### 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

Let
    $f\colon [a,b]\longrightarrow \mathbb {R}$
be a continuous function. Show that there exists a continuous extension
    ${\tilde {f}}\colon \mathbb {R} \longrightarrow \mathbb {R}$
of $f$.

### Exercise

Let $T\subseteq \mathbb {R}$ be a finite subset and let
    $f\colon T\longrightarrow \mathbb {R}$
be a function. Show that $f$ is continuous.

### Exercise

Show that there exists a continuous function
    $f\colon \mathbb {R} \longrightarrow \mathbb {R}$
such that $f$ obtains on every interval of the form $[0,\delta ]$ with $\delta >0$ positive as well as negative values.
Is it possible to draw such a function? See also Exercise 16.25 .

### 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

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

Determine the limit of the sequence
    $x_{n}={\sqrt {\frac {7n^{2}-4}{3n^{2}-5n+2}}},\,n\in \mathbb {N} .$

### Exercise

The sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is recursively defined by $x_{0}=1$ and
    $x_{n+1}={\sqrt {x_{n}+1}}\,.$
Show that this sequence converges and determine its limit.

### Exercise

Prove directly the computing rules from Lemma 10.6
(without referring to the sequence crtierion).

### Exercise

Show that the function
    $f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto {\frac {2x^{7}-3x\vert {6x^{3}-11}\vert }{\vert {3x-5}\vert +\vert {4x^{3}-5x+1}\vert }},$
is continuous.

### Exercise

Give an example for a continuous function $f\colon \mathbb {R} _{\geq 0}\rightarrow \mathbb {R} _{\geq 0}$ and an absolutely convergent real series $\sum _{k=0}^{\infty }a_{k}$ with $a_{k}\geq 0$ such that the series $\sum _{k=0}^{\infty }f(a_{k})$ does not converge.

### Exercise

Let $a\in \mathbb {R}$ and let $f,g,h\colon \mathbb {R} \rightarrow \mathbb {R}$ be functions. Suppose that $g$ and $h$ are continuous in $a$, that $g(a)=f(a)=h(a)$ holds and suppose further that $g(x)\leq f(x)\leq h(x)$ holds for all $x\in \mathbb {R}$. Show that also $f$ is continuous in $a$.

### Exercise

Determine the limit of the rational function
    ${\frac {x-1}{x^{2}-1}}$
in the point $a=1$.

### Exercise

Let $T\subseteq \mathbb {R}$ denote a subset and $a\in \mathbb {R}$ a point. Let
    $f\colon T\longrightarrow \mathbb {R}$
be a function and $b\in \mathbb {R}$ a point. Show that the following statements are equivalent.
* We have
    $\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.$
* For every sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ in $T$ which converges to $a$, also the image sequence ${\left(f(x_{n})\right)}_{n\in \mathbb {N} }$ converges to $b$.

### Exercise

Let
    $T={\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\subseteq \mathbb {R} \,$
be the set of the unit fractions and let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ denote a real sequence. Let $b\in \mathbb {R}$ and $D=T\cup \{0\}$. Show that the following properties are equivalent.
* The sequence converges to $b$.
* The function given by has a limit $\operatorname {lim} _{x\rightarrow 0}\,f(x)=b$.
    $f\colon T\longrightarrow \mathbb {R}$
    $f{\left({\frac {1}{n}}\right)}=x_{n}\,$
* The function given by and ${\tilde {f}}(0)=b$ is continuous.
    ${\tilde {f}}\colon D\longrightarrow \mathbb {R}$
    ${\tilde {f}}{\left({\frac {1}{n}}\right)}=x_{n}\,$
    Hand-in-exercises

### Exercise (3 marks)

For the function
    $f(x)=x^{3}+5x^{2}-3x+2\,$
and the point $a=3$, determine for $\epsilon ={\frac {1}{100}}$ an explicite $\delta >0$ such that
    $d(x,a)\leq \delta \,$
implies the estimate
    $d(f(x),f(a))\leq \epsilon \,.$

### 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)

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)

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 (4 marks)

Determine the limit of the rational function
    ${\frac {2x^{3}+3x^{2}-1}{x^{3}-x^{2}+x+3}}$
in the point $a=-1$.