## Mathematical Analysis and Applications: An IntroductionStarting with a discussion of Real Numbers and Functions, this text introduces standard topics of Differential and Integral Calculus together with their Applications such as Differential Equations, Numerical Analysis, Approximation Methods. |

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### Contents

TAYLORS THEOREM APPROXIMATION | 11 |

INDUCTION PRINCIPLE | 21 |

REAL NUMBER SYSTEM | 33 |

SETS AND FUNCTIONS | 49 |

CONTINUOUS FUNCTIONS I | 67 |

SEQUENCES | 73 |

CONTINUOUS FUNCTIONS | 104 |

INFINITE SERIES | 131 |

DIFFERENTIABLE FUNCTIONS | 157 |

MEAN VALUE THEOREMS | 171 |

NOTE ON PART III | 224 |

RIEMANN INTEGRATION II | 301 |

RIEMANN INTEGRATION III | 313 |

INDEX | 338 |

### Common terms and phrases

addition apply approximation argument axiom bounded called Cauchy Chapter choose clear closed compact Consider constant continuous function converges Corollary defined definition denoted derivable differentiable distance domain earlier element equal equation example Exercises exists expression f is continuous fact finite fixed function f Further geometric given gives graph Hence holds increasing induction infinite integral interval inverse irrational keep least limit point Mathematics Mean Value Theorem means method monotone namely natural Note obtained open set particular polynomial positive integer power series problems PROOF properties Proposition proved radius of convergence rational real number respectively result Riemann integral rule satisfies sequence side Similarly simple solution solving statement subset Suppose Theorem true write zero