This elementary presentation exposes readers to both the process of rigor and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim is to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Each chapter begins with the discussion of some motivating examples and concludes with a series of questions.
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Algebraic Limit Theorem analysis arbitrary argue argument assume Axiom of Completeness Cantor set Cauchy Criterion Cauchy sequence Chapter closed interval closed sets compact sets compute conclude construct contains continuous functions convergent sequence converges uniformly Definition derivative differentiable function Dirichlet's function discontinuities discussion domain element endpoints equation example Exercises Exercise f is continuous f is integrable f uniformly fact finite number fn(x follows function f functional limits Given implies infinite series intersection irrational numbers least upper bound Let f limit point lower sums mathematical Mean Value Theorem measure zero metric space natural numbers neighborhood Nested Interval nonempty open interval open sets partial sums pointwise polynomials power series prove rational numbers real numbers result Riemann integral rigorous satisfying Section sequence xn series converges set A C R Show sin(nx statement subinterval subset tagged partition Taylor series uncountable uniform convergence uniformly continuous Weierstrass
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Capture Dynamics and Chaotic Motions in Celestial Mechanics: With ...
Limited preview - 2004