# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 18

*Exercises*

Determine the Riemann sum over of the staircase function

a) Subdivide the interval in six subintervals of equal length.

b) Determine the Riemann sum of the staircase function on , which takes alternately the values and on the subdivision constructed in a).

Give an example of a function which assumes only finitely many values, but is not a staircase function.

Let

be two staircase functions. Show that also

- ,
- ,
- ,
- ,

are staircase function.

Let

be a staircase function and let

be a function. Prove that the composite is also a staircase function.

Give an example of a continuous function

and a staircase function

such that the composite is not a staircase function.

Determine the definite integral

explicitly with upper and lower staircase functions.

Determine the definite integral

explicitly with upper and lower staircase functions.

Determine the definite integral

explicitly with upper and lower staircase functions.

Show (without using primitive functions)

We consider the function

- Determine the area of the maximal lower staircase function for for an interval partitions of the form in dependence on .
- Determine between and , for which the area of the lower maximal staircase function for for an interval partition given by is maximal. What is the value of the corresponding area?

In the situation of the preceding exercise, a natural question is how the best staircase function looks like if we allow a finer subdivision, say with two points in between. We will deal with this question in the second semester, see
exercise *****.

Let be a compact interval and let

be a function. Consider a sequence of staircase functions such that and a sequence of staircase functions such that . Assume that the two Riemann sums corresponding to the sequences converge and that their limits coincide. Prove that is Riemann-integrable and that

Let be a bounded interval and let denote a continuous function which is bounded from below. Suppose that the supremum over all staircase integrals for the equidistant lower staircase functions exists. Show that then the supremum for all staircase integrals for lower staircase functions (that is, the lower integral) exists and coincides with the supremum first mentioned.

Let be a compact interval. Prove that is Riemann-integrable if and only if there is a partition

such that the restrictions

are Riemann-integrable.

Let be a compact interval and let be two Riemann-integrable functions. Prove the following statements.

- If
for all
,
then
- If
for all
,
then
- We have
- For we have .

Let be a compact interval and let be a Riemann-integrable function. Prove that

Let denote a compact interval and let denote Riemann-integrable functions. Show that also is Riemann-integrable.

Let be a compact interval and let be two Riemann-integrable functions. Prove that is also Riemann-integrable.

*The Christmas exercise for the whole family*

Which construction principle is behind the sequence

(Some people claim that this exercise is for primary school children very easy and for mathematicians quite hard.).

*Hand-in-exercises*

### Exercise (2 marks)

Let

be two staircase functions. Prove that is also a staircase function.

### Exercise (4 marks)

Determine the definite integral

as a function of and explicitly with lower and upper staircase functions.

### Exercise (4 marks)

Determine the definite integral

explicitly with upper and lower staircase functions.

### Exercise (3 marks)

Prove that for the function

neither the lower nor the upper integral exist.

### Exercise (6 marks)

Prove that for the function

the lower integral exists, but the upper integral does not exist.

Hint: Use
Exercise 9.7
.

### Exercise (5 marks)

Let be a compact interval and let

be a monotone function. Prove that is Riemann-integrable.

### Exercise (4 marks)

We consider the mapping

which is described in Exercise 18.19 (the natural numbers are given as finite sequences in the decimal system).

- Is increasing?
- Is surjective?
- Is injective?
- Does have a fixed point?

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