闭区间上连续函数基本性质证明的讨论

闭区间上连续函数基本性质证明的讨论

摘  要

闭区间上连续函数的整体性质是建立在实数完备性理论的基础之上的，而实数的完备性可以从不同的角度去刻划和描述，因此就产生了多种不同的证明闭区间上连续函数性质的方法。本文分别应用实数完备性基本定理如确界原理，区间套定理，聚点定理，有限覆盖定理和单调有界定理证明了闭区间上连续函数的3个基本性质，在应用某1实数完备性定理进行证明时，基本上没有直接应用其他完备性定理，这是本文证明的1个特点。

关键词：连续函数，闭区间，最大、最小值定理，介值性定理，1致连续性定理，完备性定理。

Abstract

    Continuous function at closed interval’s global properties was based on real number’s completeness theory, which can describe in many kinds. So there are several methods to prove it. Letterpress was introduce real number’s completeness theory such as mum principle, theorem of nested interval, theorem of accumulation, theorem of finite covering and theorem of monotonic bounded to prove it. We use only one theory to prove it.

Key words: Continuous function, closed interval, maximum-minimum theorem, intermediate value theorem, uniform continuity theorem, completeness theorem.