## Introduction to real analysisIn 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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Page 160

Robert Gardner Bartle, Donald R. Sherbert. (b) The function f + g is

at c, and (4) (f + g)\c) = f'(c) + g'(c). (c) (Product Rule) The function fg is

then the function f /g is

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) + ...

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 shallprove (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) + ...

Page 162

Therefore / is

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

Chain Rule. If / is

Rule states that ...

Therefore / is

**differentiable**at c and /'(c) = <p(c). Q.E.D. To illustrateCaratheodory'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 theChain Rule. If / is

**differentiable**at c and g is**differentiable**at / (c), then the ChainRule states that ...

Page 167

Show that f(x) := jc1/3, x e R, is not

(a), (b). 4. Let / : R -* R be defined by /(x) := a:2 for x rational, /(jc) := 0 for x

irrational. Show that / is

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 /'

Show that g(x) ...

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 andsimplify: 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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User Review - dwarfplanet9 - LibraryThingThis 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

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User Review - ssd7 - LibraryThingIntroduction 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

### Contents

PRELIMINARIES | 1 |

THE REAL NUMBERS | 22 |

SEQUENCES AND SERIES | 52 |

Copyright | |

11 other sections not shown

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### Common terms and phrases

5-fine partition absolutely convergent apply approximate arbitrary belongs bijection calculation Cauchy sequence cluster point compact conclude continuous functions Convergence Theorem convergent sequence converges uniformly cosx countable defined Definition Let denote derivative differentiable divergent elements endpoints establish Exercises for Section f is continuous finite number follows from Theorem forx function f Fundamental Theorem gauge given Hence implies increasing sequence infinite inverse Lebesgue integrable let f Let f(x lim(xn limit Mathematical Induction Mean Value Theorem metric space monotone natural number neighborhood nonempty Note obtain open interval open set partial sums polynomial properties prove rational number reader real numbers result Riemann integrable satisfy sequence of real sequence xn sinx Squeeze Theorem statement step function strictly increasing subintervals subset Suppose supremum tagged partition Taylor's Theorem Theorem Let Triangle Inequality TV[a uniform convergence uniformly continuous upper bound whence it follows