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

*Exercises*

Show that a linear function

is continuous.

Let be a subset, a function and a point. Show that the following properties are equivalent.

- is continuous in .
- For every
,
there exists some
such that for
the estimate

holds.

- For every
,
there exists some
such that for
the estimate

holds.

Prove that the function

is continuous.

Prove that the function

is continuous.

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

Let

Show that for all , the following relation holds: If

then

For the function

and the point , determine for an explicit such that

implies the estimate

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 and let

be continuous functions with

Show that there exists some such that

holds for all .

Let be a continuous function. Show the following statements.

- The function is uniquely determined by its values on .
- The value is determined by the values , .
- If for all
,
the estimate
holds, then also

holds.

Let be real numbers and let

and

be continuous functions such that . Prove that the function

such that

is also continuous.

Let

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

of .

Let be a finite subset and let

be a function. Show that is continuous.

Show that there exists a continuous function

such that obtains on every interval of the form with positive as well as negative values.

Is it possible to draw such a function? See also Exercise 16.25 .

Compute the limit of the sequence

for .

Prove that the function

defined by

is only at the zero point continuous.

Determine the limit of the sequence

The sequence is recursively defined by and

Show that this sequence converges and determine its limit.

Prove directly the computing rules from Lemma 10.6 (without referring to the sequence crtierion).

Show that the function

is continuous.

Give an example for a continuous function and an absolutely convergent real series with such that the series does not converge.

Let and let be functions. Suppose that and are continuous in , that holds and suppose further that holds for all . Show that also is continuous in .

Determine the limit of the rational function

in the point .

Let denote a subset and a point. Let

be a function and a point. Show that the following statements are equivalent.

- We have
- For every sequence in which converges to , also the image sequence converges to .

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

Let

be the set of the unit fractions and let denote a real sequence. Let and . Show that the following properties are equivalent.

- The sequence converges to .
- The function
given by

has a limit .

- The function
given by

and is continuous.

*Hand-in-exercises*

### Exercise (3 marks)

For the function

and the point , determine for an explicite such that

implies the estimate

### Exercise (2 marks)

We consider the function

Determine the points where is continuous.

### Exercise (3 marks)

Prove that the function defined by

is for no point continuous.

### Exercise (3 marks)

Compute the limit of the sequence

where

### Exercise (4 marks)

Determine the limit of the rational function

in the point .

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