## Advanced analysis: on the real lineThe goal of this book is to provide an extensive collection of results which generalize classical real analysis. Besides discussing density, approximate continuity, and approximate derivatives in detail, culminating with the Denjoy-Saks-Young Theorem, the authors also present an interesting example due to Ruziewicz on an infinite number of functions with the same derivative (not everywhere finite) but the difference of any two is not a constant and Sierpinski's theorem on the extension of approximate continuity to nonmeasurable functions. There is also a chapter on monotonic functions and one dealing with the Tonelli-Goldowsky result of the weakening of the hypotheses on a function f such that f'r >- |

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### Contents

Preliminaries | 1 |

Monotone Functions | 19 |

Density and Approximate Continuity | 37 |

Copyright | |

9 other sections not shown

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### Common terms and phrases

a,ft absolutely continuous function approximate derivative approximately continuous arbitrary Baire class bounded variation BV[a BVloc Cantor set Cantor ternary set Cantor-like set chapter closed interval co(x construction contains continuous at x0 contradiction converges Corollary countable D+f(x Darboux property Definition denote denumerable set derived numbers differentiable Dini derivatives discontinuous endpoints equal everywhere Example Exercise exists finite number function defined function f function of bounded Hausdorff Hence implies increasing function inequality infinite Lebesgue measure Lemma length Let f Let x0 measurable function measurable set measure zero metrically dense monotone function null sets open interval open set outer measure partition point of density points x e positive integer positive numbers previous theorem Proof prove rational numbers real number real-valued function segment sequence set of measure set of points singular function Stieltjes integral strictly increasing subinterval subset summable Suppose

### References to this book

Topological Fixed Point Principles for Boundary Value Problems J. Andres,Lech Górniewicz No preview available - 2003 |