閉區(qū)間上連續(xù)函數(shù)基本性質(zhì)證明的討論

閉區(qū)間上連續(xù)函數(shù)基本性質(zhì)證明的討論

閉區(qū)間上連續(xù)函數(shù)基本性質(zhì)證明的討論

摘  要

閉區(qū)間上連續(xù)函數(shù)的整體性質(zhì)是建立在實數(shù)完備性理論的基礎(chǔ)之上的，而實數(shù)的完備性可以從不同的角度去刻劃和描述，因此就產(chǎn)生了多種不同的證明閉區(qū)間上連續(xù)函數(shù)性質(zhì)的方法。本文分別應(yīng)用實數(shù)完備性基本定理如確界原理，區(qū)間套定理，聚點定理，有限覆蓋定理和單調(diào)有界定理證明了閉區(qū)間上連續(xù)函數(shù)的3個基本性質(zhì)，在應(yīng)用某1實數(shù)完備性定理進(jìn)行證明時，基本上沒有直接應(yīng)用其他完備性定理，這是本文證明的1個特點。

關(guān)鍵詞：連續(xù)函數(shù)，閉區(qū)間，最大、最小值定理，介值性定理，1致連續(xù)性定理，完備性定理。

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.

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