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

[Math Amateur]

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 $$\epsilon \gt 0$$ is arbitrary, we conclude that the right hand limit is given by

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

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

BUT ... if the expression is correct, can someone please explain how to interpret it ... ?
Peter

 

[Opalg]

In the above proof from Bartle and Sherbert we read the following:

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

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

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

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.

 

[Math Amateur]

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