Author | Berberian, Sterling K. author |
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Title | A First Course in Real Analysis [electronic resource] / by Sterling K. Berberian |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1994 |
Connect to | http://dx.doi.org/10.1007/978-1-4419-8548-4 |
Descript | XI, 240 p. online resource |
1 Axioms for the Field ? of Real Numbers -- ยง1.1. The field axioms -- ยง1.2. The order axioms -- ยง1.3. Bounded sets, LUB and GLB -- ยง1.4. The completeness axiom (existence of LUBโs) -- 2 First Properties of ? -- ยง2.1. Dual of the completeness axiom (existence of GLBโs) -- ยง2.2. Archimedean property -- ยง2.3. Bracket function -- ยง2.4. Density of the rationals -- ยง2.5. Monotone sequences -- ยง2.6. Theorem on nested intervals -- ยง2.7. Dedekind cut property -- ยง2.8. Square roots -- ยง2.9. Absolute value -- 3 Sequences of Real Numbers, Convergence -- ยง3.1. Bounded sequences -- ยง3.2. Ultimately, frequently -- ยง3.3. Null sequences -- ยง3.4. Convergent sequences -- ยง3.5. Subsequences, Weierstrass-Bolzano theorem -- ยง3.6. Cauchyโs criterion for convergence -- ยง3.7. limsup and liminf of a bounded sequence -- 4 Special Subsets of ? -- ยง4.1. Intervals -- ยง4.2. Closed sets -- ยง4.3. Open sets, neighborhoods -- ยง4.4. Finite and infinite sets -- ยง4.5. Heine-Borel covering theorem -- 5 Continuity -- ยง5.1. Functions, direct images, inverse images -- ยง5.2. Continuity at a point -- ยง5.3. Algebra of continuity -- ยง5.4. Continuous functions -- ยง5.5. One-sided continuity -- ยง5.6. Composition -- 6 Continuous Functions on an Interval -- ยง6.1. Intermediate value theorem -- ยง6.2. nโth roots -- ยง6.3. Continuous functions on a closed interval -- ยง6.4. Monotonic continuous functions -- ยง6.5. Inverse function theorem -- ยง6.6. Uniform continuity -- 7 Limits of Functions -- ยง7.1. Deleted neighborhoods -- ยง7.2. Limits -- ยง7.3. Limits and continuity -- ยง7.4. ?,?characterization of limits -- ยง7.5. Algebra of limits -- 8 Derivatives -- ยง8.1. Differentiability -- ยง8.2. Algebra of derivatives -- ยง8.3. Composition (Chain Rule) -- ยง8.4. Local max and min -- ยง8.5. Mean value theorem -- 9 Riemann Integral -- ยง9.1. Upper and lower integrals: the machinery -- ยง9.2. First properties of upper and lower integrals -- ยง9.3. Indefinite upper and lower integrals -- ยง9.4. Riemann-integrable functions -- ยง9.5. An application: log and exp -- ยง9.6. Piecewise pleasant functions -- ยง9.7. Darbouxโs theorem -- ยง9.8. The integral as a limit of Riemann sums -- 10 Infinite Series -- ยง10.1. Infinite series: convergence, divergence -- ยง10.2. Algebra of convergence -- ยง10.3. Positive-term series -- ยง10.4. Absolute convergence -- 11 Beyond the Riemann Integral -- ยง11.1 Negligible sets -- ยง11.2 Absolutely continuous functions -- ยง11.3 The uniqueness theorem -- ยง11.4 Lebesgueโs criterion for Riemann-integrability -- ยง11.5 Lebesgue-integrable functions -- ยงA.1 Proofs, logical shorthand -- ยงA.2 Set notations -- ยงA.3 Functions -- ยงA.4 Integers -- Index of Notations