## The Way of AnalysisThe Way of Analysis gives a thorough account of real analysis in one or several variables, from the construction of the real number system to an introduction of the Lebesgue integral. The text provides proofs of all main results, as well as motivations, examples, applications, exercises, and formal chapter summaries. Additionally, there are three chapters on application of analysis, ordinary differential equations, Fourier series, and curves and surfaces to show how the techniques of analysis are used in concrete settings. |

### Contents

Preliminaries | 1 |

Topology of the Real Line | 3 |

Differential Calculus | 5 |

Transcendental Functions | 8 |

Implicit Functions Curves and Surfaces | 13 |

Concepts of Continuity | 14 |

Construction of the Real Number System | 25 |

Exercises | 34 |

Euclidean Space and Metric Spaces | 355 |

Continuous Functions on Compact Domains | 386 |

30 | 412 |

Differential Calculus in Euclidean Space | 445 |

Ordinary Differential Equations | 459 |

Fourier Series | 522 |

The Lebesgue Integral | 624 |

Multiple Integrals | 698 |

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### Common terms and phrases

1/m there exists 1/n implies absolutely convergent approximation assume axiom of Archimedes Cauchy criterion Cauchy sequence Cauchy sums closed set compact set complex numbers construction contains continuous function Dedekind cut definition denote derivative difference quotient differentiable at xo discontinuities domain endpoints equivalence class error 1/n exercise f(xo fact finite number fn(x formula function f ƒ and g ƒ is continuous g(xo given any error implies f(x intuitive lemma Let f limit limit-point limsup mathematics mean value theorem metric space monotone increasing natural number neighborhood of xo obtain open interval open set Osc(f partition point xo polynomial positive power series proof properties prove rational numbers real number system Riemann integrable satisfies sequence of rationals sequence x1 shown in Figure statement subintervals subset sup and inf Taylor's theorem triangle inequality uniform convergence upper bound x-xo y₁ zero