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

Exercises

Exercise

Show that a linear function

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

is continuous.

Exercise

Let ${\displaystyle {}D\subseteq \mathbb {R} }$ be a subset, ${\displaystyle {}f\colon D\rightarrow \mathbb {R} }$ a function and ${\displaystyle {}a\in D}$ a point. Show that the following properties are equivalent.

1. ${\displaystyle {}f}$ is continuous in ${\displaystyle {}a}$.
2. For every ${\displaystyle {}n\in \mathbb {N} _{+}}$, there exists some ${\displaystyle {}m\in \mathbb {N} _{+}}$ such that for
${\displaystyle {}\vert {x-a}\vert \leq {\frac {1}{m}}\,}$

the estimate

${\displaystyle {}\vert {f(x)-f(a)}\vert \leq {\frac {1}{n}}\,}$

holds.

3. For every ${\displaystyle {}s\in \mathbb {N} }$, there exists some ${\displaystyle {}r\in \mathbb {N} }$ such that for
${\displaystyle {}\vert {x-a}\vert \leq {\frac {1}{10^{r}}}\,}$

the estimate

${\displaystyle {}\vert {f(x)-f(a)}\vert \leq {\frac {1}{10^{s}}}\,}$

holds.

Exercise

Prove that the function

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

is continuous.

Exercise

Prove that the function

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

is continuous.

Exercise

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

Exercise

Let

${\displaystyle {}f(x)=2x^{3}-4x+5\,.}$

Show that for all ${\displaystyle {}x\in \mathbb {R} }$, the following relation holds: If

${\displaystyle {}\vert {x-3}\vert \leq {\frac {1}{800}}\,,}$

then

${\displaystyle {}\vert {f(x)-f(3)}\vert \leq {\frac {1}{10}}\,.}$

Exercise

For the function

${\displaystyle {}f(x)=2x^{3}-4x^{2}+x-6\,}$

and the point ${\displaystyle {}a=1}$, determine for ${\displaystyle {}\epsilon ={\frac {1}{10}}}$ an explicit ${\displaystyle {}\delta >0}$ such that

${\displaystyle {}d(x,a)\leq \delta \,}$

implies the estimate

${\displaystyle {}d(f(x),f(a))\leq \epsilon \,.}$

Exercise

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

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

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

Exercise

Let ${\displaystyle {}a\in \mathbb {R} }$ and let

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

be continuous functions with

${\displaystyle {}f(a)>g(a)\,.}$

Show that there exists some ${\displaystyle {}\delta >0}$ such that

${\displaystyle {}f(x)>g(x)\,}$

holds for all ${\displaystyle {}x\in [a-\delta ,a+\delta ]}$.

Exercise

Let ${\displaystyle {}f\colon \mathbb {R} \rightarrow \mathbb {R} }$ be a continuous function. Show the following statements.

1. The function ${\displaystyle {}f}$ is uniquely determined by its values on ${\displaystyle {}\mathbb {Q} }$.
2. The value ${\displaystyle {}f(a)}$ is determined by the values ${\displaystyle {}f(x)}$, ${\displaystyle {}x\neq a}$.
3. If for all ${\displaystyle {}x, the estimate
${\displaystyle {}f(x)\leq c\,}$

holds, then also

${\displaystyle {}f(a)\leq c\,}$

holds.

Exercise

Let ${\displaystyle {}a be real numbers and let

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

and

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

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

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

such that

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

is also continuous.

Exercise

Let

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

be a continuous function. Show that there exists a continuous extension

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

of ${\displaystyle {}f}$.

Exercise

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

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

be a function. Show that ${\displaystyle {}f}$ is continuous.

Exercise

Show that there exists a continuous function

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

such that ${\displaystyle {}f}$ obtains on every interval of the form ${\displaystyle {}[0,\delta ]}$ with ${\displaystyle {}\delta >0}$ 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

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

for ${\displaystyle {}n\rightarrow \infty }$.

Exercise

Prove that the function

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

defined by

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

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

Exercise

Determine the limit of the sequence

${\displaystyle x_{n}={\sqrt {\frac {7n^{2}-4}{3n^{2}-5n+2}}},\,n\in \mathbb {N} .}$

Exercise

The sequence ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ is recursively defined by ${\displaystyle {}x_{0}=1}$ and

${\displaystyle {}x_{n+1}={\sqrt {x_{n}+1}}\,.}$

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

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto {\frac {2x^{7}-3x\vert {6x^{3}-11}\vert }{\vert {3x-5}\vert +\vert {4x^{3}-5x+1}\vert }},}$

is continuous.

Exercise

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

Exercise

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

Exercise

Determine the limit of the rational function

${\displaystyle {\frac {x-1}{x^{2}-1}}}$

in the point ${\displaystyle {}a=1}$.

Exercise

Let ${\displaystyle {}T\subseteq \mathbb {R} }$ denote a subset and ${\displaystyle {}a\in \mathbb {R} }$ a point. Let

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

be a function and ${\displaystyle {}b\in \mathbb {R} }$ a point. Show that the following statements are equivalent.

1. We have
${\displaystyle {}\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.}$
2. For every sequence ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ in ${\displaystyle {}T}$ which converges to ${\displaystyle {}a}$, also the image sequence ${\displaystyle {}{\left(f(x_{n})\right)}_{n\in \mathbb {N} }}$ converges to ${\displaystyle {}b}$.

Hint: This is proved similarly to the sequence criterion for continuity.

Exercise

Let

${\displaystyle {}T={\left\{{\frac {1}{n}}\mid n\in \mathbb {N} _{+}\right\}}\subseteq \mathbb {R} \,}$

be the set of the unit fractions and let ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ denote a real sequence. Let ${\displaystyle {}b\in \mathbb {R} }$ and ${\displaystyle {}D=T\cup \{0\}}$. Show that the following properties are equivalent.

1. The sequence converges to ${\displaystyle {}b}$.
2. The function
${\displaystyle f\colon T\longrightarrow \mathbb {R} }$

given by

${\displaystyle {}f{\left({\frac {1}{n}}\right)}=x_{n}\,}$

has a limit ${\displaystyle {}\operatorname {lim} _{x\rightarrow 0}\,f(x)=b}$.

3. The function
${\displaystyle {\tilde {f}}\colon D\longrightarrow \mathbb {R} }$

given by

${\displaystyle {}{\tilde {f}}{\left({\frac {1}{n}}\right)}=x_{n}\,}$

and ${\displaystyle {}{\tilde {f}}(0)=b}$ is continuous.

Hand-in-exercises

Exercise (3 marks)

For the function

${\displaystyle {}f(x)=x^{3}+5x^{2}-3x+2\,}$

and the point ${\displaystyle {}a=3}$, determine for ${\displaystyle {}\epsilon ={\frac {1}{100}}}$ an explicite ${\displaystyle {}\delta >0}$ such that

${\displaystyle {}d(x,a)\leq \delta \,}$

implies the estimate

${\displaystyle {}d(f(x),f(a))\leq \epsilon \,.}$

Exercise (2 marks)

We consider the function

${\displaystyle {}f(x)={\begin{cases}1{\text{ for }}x\leq -1\,,\\x^{2}{\text{ for }}-1

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

Exercise (3 marks)

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

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

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

Exercise (3 marks)

Compute the limit of the sequence

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

where

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

Exercise (4 marks)

Determine the limit of the rational function

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

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