## Complexity Theory of Real FunctionsStarting with Cook's pioneering work on NP-completeness in 1970, polynomial complexity theory, the study of polynomial-time com putability, has quickly emerged as the new foundation of algorithms. On the one hand, it bridges the gap between the abstract approach of recursive function theory and the concrete approach of analysis of algorithms. It extends the notions and tools of the theory of computability to provide a solid theoretical foundation for the study of computational complexity of practical problems. In addition, the theoretical studies of the notion of polynomial-time tractability some times also yield interesting new practical algorithms. A typical exam ple is the application of the ellipsoid algorithm to combinatorial op timization problems (see, for example, Lovasz [1986]). On the other hand, it has a strong influence on many different branches of mathe matics, including combinatorial optimization, graph theory, number theory and cryptography. As a consequence, many researchers have begun to re-examine various branches of classical mathematics from the complexity point of view. For a given nonconstructive existence theorem in classical mathematics, one would like to find a construc tive proof which admits a polynomial-time algorithm for the solution. One of the examples is the recent work on algorithmic theory of per mutation groups. In the area of numerical computation, there are also two tradi tionally independent approaches: recursive analysis and numerical analysis. |

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### Contents

1 | |

Basics in Discrete Complexity Theory | 12 |

Computational Complexity of Real | 40 |

Maximization | 71 |

Roots and Inverse Functions | 107 |

Measure and Integration | 159 |

Differentiation | 190 |

Ordinary Differential Equations | 215 |

Approximation by Polynomials | 247 |

An Optimization Problem in Control | 274 |

290 | |

301 | |

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algorithm binary expansion bits bounded breakpoints claim completes the proof complexity classes complexity theory computable in polynomial computable real functions computable real numbers converges Corollary define definition denote the class discrete complexity theory dyadic rational equation 7.1 error f e Paſo.1 f is polynomial-time f satisfies finite func function f function for f halts implies input integer interval inverse function LCTEAM left cut Lemma Lipschitz condition log-space computable maps maximum points modulus function modulus of continuity node nondeterministic TM Note notion NP-complete one-to-one function one-way function oracle machine oracle TM output P/poly piecewise linear functions polynomial function polynomial modulus polynomial-time approximable polynomial-time computable function polynomial-time computable real problem proof of Theorem PSPACE putable rational number recursive function recursively approximable recursively open solution sparse sets string tally set tion weakly polynomial-time computable