# The Riemann and Darboux Integrals .... Browder, Theorem 5.10 .... ....

• MHB
• Math Amateur
In summary: S}(f,\pi')\right| < \frac{\epsilon}{2}$$Now, let's consider the partition \pi = \pi' \cup \{x_0\} where x_0 is any point in the interval [a,b]. This means that we add one point to the partition \pi', making it a finer partition. This finer partition will result in a smaller difference between the upper and lower sums, since the supremum and infimum of f on each subinterval will be closer together.Thus, we have$$\overline{S}(f,\pi) - \
Math Amateur
Gold Member
MHB
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ...

I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof read as follows:View attachment 9473
View attachment 9474

At the start of the above proof by Browder we read the following:

" ... ... The necessity of the condition is immediate from the definition of the integral ... ... "Can someone please help me to rigorously demonstrate the necessity of the condition ...

-------------------------------------------------------------------------------------------------------------------

Note: I am assuming that proving "the necessity of the condition is proving the following:$$\displaystyle \int_a^b f \text{ exists } \Longrightarrow$$ ... for every $$\displaystyle \epsilon \gt 0 \ \exists \$$ a partition $$\displaystyle \pi$$ of $$\displaystyle [a, b]$$ such that $$\displaystyle \overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon$$ ... ...

-------------------------------------------------------------------------------------------------------------------

Help will be much appreciated ...

Peter
==========================================================================================Note: It may help MHB readers of the above post to have access to Browder's notation, definitions and theorems on Riemann integration preliminary to Theorem 5.10 ... hence i am providing access to the same ... as follows:
View attachment 9475
View attachment 9476
View attachment 9477
View attachment 9478
Hope that helps ...

Peter

#### Attachments

• Browder - 1 - Theorem 5.10 ... PART 1 ... .png
22.9 KB · Views: 95
• Browder - 2 - Theorem 5.10 ... PART 2 ... .png
4.8 KB · Views: 87
• Browder - 1 - Start of 5.1 - Relevant Defns & Propns ... PART 1 ... .png
18.1 KB · Views: 96
• Browder - 2 - Start of 5.1 - Relevant Defns & Propns ... PART 2 ... .png
57.5 KB · Views: 99
• Browder - 3 - Start of 5.1 - Relevant Defns & Propns ... PART 3 ... .png
44.6 KB · Views: 88
• Browder - 4 - Start of 5.1 - Relevant Defns & Propns ... PART 4 ... .png
43.2 KB · Views: 89
Last edited:
Peter said:
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.1 Riemann Sums ... ...

I need some help in understanding the proof of Theorem 5.10 ...Theorem 5.10 and its proof read as follows:At the start of the above proof by Browder we read the following:

" ... ... The necessity of the condition is immediate from the definition of the integral ... ... "Can someone please help me to rigorously demonstrate the necessity of the condition ...

-------------------------------------------------------------------------------------------------------------------

Note: I am assuming that proving "the necessity of the condition is proving the following:$$\displaystyle \int_a^b f \text{ exists } \Longrightarrow$$ ... for every $$\displaystyle \epsilon \gt 0 \ \exists \$$ a partition $$\displaystyle \pi$$ of $$\displaystyle [a, b]$$ such that $$\displaystyle \overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon$$ ... ...

-------------------------------------------------------------------------------------------------------------------

Help will be much appreciated ...

Peter
==========================================================================================Note: It may help MHB readers of the above post to have access to Browder's notation, definitions and theorems on Riemann integration preliminary to Theorem 5.10 ... hence i am providing access to the same ... as follows:

Hope that helps ...

Peter
I have been reflecting on my problem in the above post and now give my attempted proof of

$$\displaystyle \int_a^b f \text{ exists } \Longrightarrow$$ ... for every $$\displaystyle \epsilon \gt 0 \ \exists \$$ a partition $$\displaystyle \pi$$ of $$\displaystyle [a, b]$$ such that $$\displaystyle \overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon$$ ... ...Proof:

Let $$\displaystyle \int_a^b f = I$$

Then

$$\displaystyle I$$ exists $$\displaystyle \Longrightarrow$$ for any $$\displaystyle \frac{ \epsilon }{2} \gt 0 \ \exists \ \pi_0$$ such that for any $$\displaystyle \pi \geq \pi_0$$ and every selection $$\displaystyle \sigma$$ associated with $$\displaystyle \pi$$ we have $$\displaystyle | s(f, \pi, \sigma ) - I | \lt \frac{ \epsilon }{2}$$Now $$\displaystyle | S(f, \pi, \sigma ) - I | \lt \frac{ \epsilon }{2}$$

implies that

$$\displaystyle - \frac{ \epsilon }{2} \lt S(f, \pi, \sigma ) - I \lt \frac{ \epsilon }{2}$$

and so, obviously, we have that

$$\displaystyle S(f, \pi, \sigma ) - I \lt \frac{ \epsilon }{2}$$ ... ... ... ... ... (1)But $$\displaystyle | S(f, \pi, \sigma ) - I | \lt \frac{ \epsilon }{2}$$

... also implies that

$$\displaystyle - \frac{ \epsilon }{2} \lt I - S(f, \pi, \sigma ) \lt \frac{ \epsilon }{2}$$

so, obviously, we have that

$$\displaystyle I - S(f, \pi, \sigma ) \lt \frac{ \epsilon }{2}$$ ... ... ... ... ... (2)

Now we also have that

$$\displaystyle \underline{S} (f, \pi) \leq S(f, \pi, \sigma ) \leq \overline{S} (f, \pi)$$ ... ... ... ... ... (3)Now (1) and (3) imply $$\displaystyle \overline{S} (f, \pi) - I \lt \frac{ \epsilon }{2}$$ ... ... ... ... ... (4)Similarly (2) and (3) imply$$\displaystyle I - \underline{S} (f, \pi) \lt \frac{ \epsilon }{2}$$ ... ... ... ... ... (5)Adding (4) and (5) gives $$\displaystyle \overline{S} (f, \pi) - \underline{S} (f, \pi) \lt \epsilon$$ ... ...
Can someone please critique the above proof and either confirm it is correct and/or point out errors or shortcomings ...

Peter

Last edited:
Hello Peter,

I am also currently reading Browder's book and I understand your confusion about the necessity of the condition in Theorem 5.10. I will try my best to explain it to you.

First of all, let's define the notation used in the theorem. The upper and lower sums, denoted by $\overline{S}(f,\pi)$ and $\underline{S}(f,\pi)$ respectively, are defined as follows:

$$\overline{S}(f,\pi) = \sum_{i=1}^n M_i \Delta x_i$$
$$\underline{S}(f,\pi) = \sum_{i=1}^n m_i \Delta x_i$$

where $M_i$ and $m_i$ are the supremum and infimum of $f$ on the $i$th subinterval $[x_{i-1},x_i]$, and $\Delta x_i$ is the length of the subinterval.

Now, let's look at the definition of the Riemann integral. A function $f$ is said to be Riemann integrable on $[a,b]$ if and only if for every $\epsilon > 0$, there exists a partition $\pi$ of $[a,b]$ such that

$$\overline{S}(f,\pi) - \underline{S}(f,\pi) < \epsilon$$

This means that for any small $\epsilon$, we can always find a partition such that the difference between the upper and lower sums is less than $\epsilon$. This is the condition that Browder is referring to in the proof of Theorem 5.10.

Now, to prove the necessity of this condition, we need to show that if the Riemann integral exists, then this condition must hold. In other words, if we can find a partition $\pi$ such that the difference between the upper and lower sums is less than any given $\epsilon$, then the Riemann integral must exist.

To prove this, we can use the definition of the Riemann integral again. Since we assume that the integral exists, we know that for any $\epsilon > 0$, there exists a partition $\pi'$ such that

$$\left|\int_a^b f(x) dx - \overline{S}(f,\pi')\right| < \frac{\epsilon}{2}$$

and

## 1. What is the Riemann integral?

The Riemann integral is a mathematical concept used to calculate the area under a curve. It is named after the mathematician Bernhard Riemann and is defined as the limit of a sum of rectangles that approximate the area under a curve. It is an important tool in calculus and is used to solve a variety of problems in physics, engineering, and economics.

## 2. What is the Darboux integral?

The Darboux integral is another method of calculating the area under a curve. It is named after the mathematician Jean Gaston Darboux and is defined as the limit of a sum of rectangles that are inscribed within the curve. It is closely related to the Riemann integral and is also used in calculus to solve problems in various fields.

## 3. What is Theorem 5.10 in Browder's book?

Theorem 5.10 in Browder's book is a mathematical theorem that states that if a function is continuous on a closed interval, then it is also integrable on that interval. This means that the Riemann and Darboux integrals will give the same result for a continuous function on a closed interval.

## 4. How is Theorem 5.10 useful in calculus?

Theorem 5.10 is useful in calculus because it allows us to determine if a function is integrable on a given interval. This is important in solving various problems in physics, engineering, and economics that involve calculating areas under curves. Additionally, it helps us understand the relationship between the Riemann and Darboux integrals.

## 5. Are there any real-life applications of Theorem 5.10?

Yes, Theorem 5.10 has many real-life applications. For example, it is used in economics to calculate the total profit or loss of a company based on their revenue function. It is also used in engineering to determine the total work done by a variable force. Additionally, it is used in physics to calculate the total energy of a system based on its force function.

Replies
6
Views
2K
Replies
2
Views
2K
Replies
2
Views
1K
Replies
4
Views
2K
Replies
14
Views
2K
Replies
2
Views
1K
Replies
2
Views
2K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
2
Views
1K