Another Question On B&S, Theorem 7.3.5 - Fundamental Theorem Of Calculus ...

In summary, Bartle and Sherbert discuss Theorem 7.3.5 and its proof. They mention that the right hand limit is given by \text{lim}_{ x \rightarrow 0+ } \frac{ F( c + h ) - F(c) }{h} = f(c)
  • #1
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I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...

I am focused on Chapter 7: The Riemann Integral ...

I need help in fully understanding yet another aspect of the proof of Theorem 7.3.5 ...Theorem 7.3.5 and its proof ... ... read as follows:View attachment 7327
In the above proof from Bartle and Sherbert we read the following:

" ... ... But, since \(\displaystyle \epsilon \gt 0\) is arbitrary, we conclude that the right hand limit is given by

\(\displaystyle \text{lim}_{ x \rightarrow 0+ } \frac{ F( c + h ) - F(c) }{h} = f(c) \)

... ... "Should this read \(\displaystyle \text{lim}_{ h \rightarrow 0+ } \frac{ F( c + h ) - F(c) }{h} = f(c)\) ... ...?

BUT ... if the expression is correct, can someone please explain how to interpret it ... ?
Peter
 
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  • #2
Peter said:
In the above proof from Bartle and Sherbert we read the following:

" ... ... But, since \(\displaystyle \epsilon \gt 0\) is arbitrary, we conclude that the right hand limit is given by

\(\displaystyle \text{lim}_{ x \rightarrow 0+ } \frac{ F( c + h ) - F(c) }{h} = f(c) \)

... ... "Should this read \(\displaystyle \text{lim}_{ h \rightarrow 0+ } \frac{ F( c + h ) - F(c) }{h} = f(c)\) ... ...?
Yet another typo! Yes, it should obviously be $\lim_{h\to0+}$.

Robert Bartle was a well-known mathematician, who specialised in writing introductory real analysis textbooks. The first edition of Introduction to Real Analysis, which he co-authored with Donald Sherbert, appeared in 1983, and this book has been popular ever since. But Bartle died in 2003, and it seems that later editions (published by Wiley) have not been prepared as carefully as the original.
 
  • #3
Opalg said:
Yet another typo! Yes, it should obviously be $\lim_{h\to0+}$.

Robert Bartle was a well-known mathematician, who specialised in writing introductory real analysis textbooks. The first edition of Introduction to Real Analysis, which he co-authored with Donald Sherbert, appeared in 1983, and this book has been popular ever since. But Bartle died in 2003, and it seems that later editions (published by Wiley) have not been prepared as carefully as the original.
Thanks Opalg ...

Peter
 

Related to Another Question On B&S, Theorem 7.3.5 - Fundamental Theorem Of Calculus ...

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a theorem in mathematics that establishes the relationship between differentiation and integration. It states that if a function is continuous on a closed interval and has an antiderivative, then the definite integral of the function over that interval can be calculated by evaluating the antiderivative at the endpoints of the interval.

How is Theorem 7.3.5 related to the Fundamental Theorem of Calculus?

Theorem 7.3.5 is a specific version of the Fundamental Theorem of Calculus that deals with definite integrals of functions that have discontinuities at a finite number of points. It provides a way to calculate the definite integral by breaking it into smaller, continuous intervals and using the Fundamental Theorem of Calculus on each interval.

What are the conditions for the Fundamental Theorem of Calculus to hold?

The Fundamental Theorem of Calculus holds if the function is continuous on the closed interval and has an antiderivative on that interval. Additionally, the upper and lower limits of integration must be within the interval and the function must have a finite number of discontinuities.

Can the Fundamental Theorem of Calculus be used to evaluate indefinite integrals?

No, the Fundamental Theorem of Calculus only applies to definite integrals. To evaluate indefinite integrals, the rules of integration such as the power rule, substitution, and integration by parts must be used.

Is the Fundamental Theorem of Calculus important in real-world applications?

Yes, the Fundamental Theorem of Calculus is an essential tool in many real-world applications, particularly in physics and engineering. It allows for the calculation of areas, volumes, and other quantities that require integration. It is also used in economics and statistics to model and analyze continuous functions.

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