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



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  .