Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 15/en/latex

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\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Show that a \definitionsverweis {linear function}{}{} \maabbeledisp {} {\R} {\R } {x} {ax } {,} is continuous.

}
{} {}




\inputexercise
{}
{

Prove that the function \maabbeledisp {} {\R } {\R } {x} { \betrag { x } } {,} is continuous.

}
{} {}




\inputexercise
{}
{

Prove that the function \maabbeledisp {} {\R_{\geq 0} } {\R_{\geq 0} } {x} { \sqrt{x} } {,} is continuous.

}
{} {}




\inputexercise
{}
{

Let
\mathl{T \subseteq \R}{} be a subset and let \maabbdisp {f} {T} {\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 $]x- a, x + a[$.

}
{} {}




\inputexercise
{}
{

Let $a < b < c$ be real numbers and let \maabbdisp {f} {[a,b]} {\R } {} and \maabbdisp {g} {[b,c]} {\R } {} be continuous functions such that $f(b) = g(b)$. Prove that the function \maabbdisp {h} {[a,c]} {\R } {} such that
\mathdisp {h(t) = f(t) {{\text{ for }}} t \leq b \text{ and } h(t) = g(t) {{\text{ for }}} t > b} { }
is also continuous.

}
{} {}




\inputexercise
{}
{

Compute the limit of the sequence
\mathdisp {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
\mathl{n \rightarrow \infty}{.}

}
{} {}




\inputexercise
{}
{

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

}
{} {}




\inputexercise
{}
{

Give an example of a continuuos function \maabbdisp {f} {\Q} {\R } {,} which takes exactly two values​​.

}
{} {}




\inputexercise
{}
{

Prove that the function \maabbdisp {f} {\R} {\R } {} defined by
\mathdisp {f(x) = \begin{cases} x ,\, \text{ if } x \in \Q \, , \\ 0,\, \text{ otherwise} \, , \end{cases}} { }
is only at the zero point $0$ continuous.

}
{} {}




\inputexercise
{}
{

Let
\mathl{T \subseteq \R}{} be a subset and let
\mathl{a \in \R}{} be a point. Let \maabb {f} {T} {\R } {} be a function and
\mathl{b \in \R}{.} Prove that the following statements are equivalent. \aufzaehlungzwei {We have
\mathdisp {\operatorname{lim}_{ x \rightarrow a } \, f(x) = b} { . }
} {For all
\mathl{\epsilon >0}{} there exists a
\mathl{\delta >0}{} such that for all
\mathl{x \in T}{} with
\mathl{d(x,a) \leq \delta}{} the inequality
\mathl{d(f(x),b) \leq \epsilon}{} holds. }

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{4}
{

We consider the function
\mathdisp {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
\mathl{x \in \R}{} where $f$ is \definitionsverweis {continuous}{}{.}

}
{} {}




\inputexercise
{4}
{

Compute the limit of the sequence
\mathdisp {b_n =2a_n^4-6 a_n^3+a_n^2-5a_n+3} { , }
where
\mathdisp {a_n = \frac{3n^3-5n^2+7}{4n^3+2n-1}} { . }

}
{} {}




\inputexercise
{3}
{

Prove that the function \maabb {f} {\R} {\R } {} defined by
\mathdisp {f(x) = \begin{cases} 1, \text{ if } x \in \Q \, , \\ 0 \text{ otherwise} \, , \end{cases}} { }
is for no point $x \in \R$ continuous.

}
{} {}




\inputexercise
{3}
{

Decide whether the sequence
\mathdisp {a_n = \sqrt{n+1} - \sqrt{n}} { }
converges and in case determine the limit.

}
{} {}




\inputexercise
{4}
{

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

}
{} {}



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