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首页> 外文期刊>Real analysis exchange >A NOTE ON MONOTONICITY THEOREMS FOR APPROXIMATELY CONTINUOUS FUNCTIONS
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A NOTE ON MONOTONICITY THEOREMS FOR APPROXIMATELY CONTINUOUS FUNCTIONS

机译：近似连续函数的单调性定理的一个注记

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摘要

We say that f(x) is approximately continuous at a if there is a measurable set V such that a is a density point of V and the restriction f|y (x) is continuous at a. If in addition f|v(x) is differentiable at a then we say that f(x) is approximately differentiable at a, and we denote the derivative by f'_(ap)(a). It is well known, Theorem 2.5 [2], that f'_(ap)(x) > 0 at every point x of an interval I implies that / is nondecreasing on I. See [2], page 107 for the proof of this result. The first part of this note is to provide a simple proof of Theorem 2.5. Since the conditions used in our proof are much weaker than those of Theorem 2.5, our Theorem 2 can be also regarded as its generalization.
机译：我们说，如果有一个可测量的集合V使得f（x）在a处近似连续，使得a是V的密度点，并且限制f | y（x）在a处连续。如果另外f | v（x）在a处是可微的，那么我们说f（x）在a处是可微的，我们用f'_（ap）（a）表示导数。众所周知，定理2.5 [2]，在间隔I的每个点x上f'_（ap）（x）> 0表示I在/​​上不递减。有关证明，请参见第2页，第107页这个结果。本说明的第一部分是提供定理2.5的简单证明。由于我们的证明所使用的条件比定理2.5的条件弱得多，因此我们的定理2也可以视为其推广。

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