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AuthorBerberian, Sterling K. author
TitleA First Course in Real Analysis [electronic resource] / by Sterling K. Berberian
ImprintNew York, NY : Springer New York : Imprint: Springer, 1994
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Descript XI, 240 p. online resource


Mathematics is the music of science, and real analysis is the Bach of mathematics. There are many other foolish things I could say about the subject of this book, but the foregoing will give the reader an idea of where my heart lies. The present book was written to support a first course in real analysis, normally taken after a year of elementary calculus. Real analysis is, roughly speaking, the modern setting for Calculus, "real" alluding to the field of real numbers that underlies it all. At center stage are functions, defined and taking values in sets of real numbers or in sets (the plane, 3-space, etc.) readily derived from the real numbers; a first course in real analysis traditionally places the emphasis on real-valued functions defined on sets of real numbers. The agenda for the course: (1) start with the axioms for the field ofreal numbers, (2) build, in one semester and with appropriate rigor, the founยญ dations of calculus (including the "Fundamental Theorem"), and, along the way, (3) develop those skills and attitudes that enable us to continue learning mathematics on our own. Three decades of experience with the exercise have not diminished my astonishment that it can be done


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 LUBโ{128}{153}s) -- 2 First Properties of ? -- ยง2.1. Dual of the completeness axiom (existence of GLBโ{128}{153}s) -- 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. Cauchyโ{128}{153}s 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. nโ{128}{153}th 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. Darbouxโ{128}{153}s 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 Lebesgueโ{128}{153}s 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

Mathematics Functions of real variables Mathematics Real Functions


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