Jump to content

Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 15

From Wikiversity



Warm-up-exercises


Show that a linear function

is continuous.



Prove that the function

is continuous.



Prove that the function

is continuous.



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 .



Let be real numbers and let

and

be continuous functions such that . Prove that the function

such that

is also continuous.


Compute the limit of the sequence

for .



Let

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



Give an example of a continuous function

which takes exactly two values​​.



Prove that the function

defined by

is only at the zero point continuous.



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 (2 marks)

We consider the function

Determine the points where is continuous.



Exercise (3 marks)

Compute the limit of the sequence

where



Exercise (3 marks)

Prove that the function defined by

is for no point continuous.



Exercise (3 marks)

Decide whether the sequence

converges and in case determine the limit.



Exercise (4 marks)

Determine the limit of the rational function

in the point .