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

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Warm-up-exercises

Exercise

Show that a linear function

is continuous.


Exercise

Prove that the function

is continuous.


Exercise

Prove that the function

is continuous.


Exercise

Let be a subset and let

be a continuous function. Let be a point such that . Prove that for all in a non-empty open interval .


Exercise

Let be real numbers and let

and

be continuous functions such that . Prove that the function

such that

is also continuous.


Exercise

Compute the limit of the sequence

for .


Exercise

Let

be a continuous function which takes only finitely many values. Prove that is constant.


Exercise

Give an example of a continuuos function

which takes exactly two values​​.


Exercise

Prove that the function
defined by
is only at the zero point continuous.


Exercise

Let be a subset and let be a point. Let be a function and . Prove that the following statements are equivalent.

  1. We have
  2. For all there exists a such that for all with the inequality holds.




Hand-in-exercises

Exercise (4 points)

We consider the function

Determine the points where is continuous.


Exercise (4 points)

Compute the limit of the sequence

where


Exercise (3 points)

Prove that the function defined by

is for no point continuous.


Exercise (3 points)

Decide whether the sequence

converges and in case determine the limit.


Exercise (4 points)

Determine the limit of the rational function

at the point .



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