Introduction to real analysis
John Wiley & Sons Canada, Limited, 2000 - Mathematics - 388 pages
In recent years, mathematics has become valuable in many areas, including economics and management science as well as the physical sciences, engineering and computer science. Therefore, this book provides the fundamental concepts and techniques of real analysis for readers in all of these areas. It helps one develop the ability to think deductively, analyze mathematical situations and extend ideas to a new context. Like the first two editions, this edition maintains the same spirit and user-friendly approach with some streamlined arguments, a few new examples, rearranged topics, and a new chapter on the Generalized Riemann Integral.
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Robert Gardner Bartle, Donald R. Sherbert. (b) The function f + g is differentiable
at c, and (4) (f + g)\c) = f'(c) + g'(c). (c) (Product Rule) The function fg is
differentiable at c, and (5) (/g)'(c) = /'(c)g(c) + /(c)g'(c). (d) (Quotient Rule) If g(c) / 0,
then the function f /g is differentiable at c, and (c)g(c) - /(c)g'(c) (6) /Voo/ We shall
prove (c) and (d), leaving (a) and (b) as exercises for the reader. (c) Let p := fg;
then for x € I, x ^ c, we have £fr) ~ p(c) = /(x)g(x) - /(c)g(c) x — c a: — c = /(x)g(x) - f(
c)g(x) + ...
Therefore / is differentiable at c and /'(c) = <p(c). Q.E.D. To illustrate
Caratheodory's Theorem, we consider the function / defined by fix) := x3 for x € R.
For c € R, we see from the factorization x3 - c3 = (x2 + cx + c2)(x - c) that <p(x) :=
x2 + cx + c2 satisfies the conditions of the theorem. Therefore, we conclude that /
is differentiable at c € R and that /'(c) = <p(c) = 3c2. We will now establish the
Chain Rule. If / is differentiable at c and g is differentiable at / (c), then the Chain
Rule states that ...
Show that f(x) := jc1/3, x e R, is not differentiable at x = 0. 3. Prove Theorem 6. 1 .3
(a), (b). 4. Let / : R -* R be defined by /(x) := a:2 for x rational, /(jc) := 0 for x
irrational. Show that / is differentiable at x = 0, and find /'(0). 5. Differentiate and
simplify: 6. Letn € N and let / : R -* R be defined by /(x) := x"forx > Oand/(x) :=0for;c
< O.For which values of n is /' continuous at 0? For which values of n is /'
differentiable at 0? 7. Suppose that / : R -> R is differentiable at c and that /(c) = 0.
Show that g(x) ...
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LibraryThing ReviewUser Review - dwarfplanet9 - LibraryThing
This book was used in my Real Analysis course. The subject would be hard to learn from this book alone, but lucky for me I had a great teacher at San Jose State University. For those trying to use the ... Read full review
LibraryThing ReviewUser Review - ssd7 - LibraryThing
Introduction to Real Analysis is easily one of my favorite mathematics textbooks. The explanation is excellent and the in-text examples are interesting. Unlike most mathematics text books I've read ... Read full review
THE REAL NUMBERS
SEQUENCES AND SERIES
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