閉區間上連續函數基本性質證明的討論
摘 要
閉區間上連續函數的整體性質是建立在實數完備性理論的基礎之上的'，而實數的完備性可以從不同的角度去刻劃和描述，因此就產生了多種不同的證明閉區間上連續函數性質的方法。本文分別應用實數完備性基本定理如確界原理，區間套定理，聚點定理，有限覆蓋定理和單調有界定理證明了閉區間上連續函數的3個基本性質，在應用某1實數完備性定理進行證明時，基本上沒有直接應用其他完備性定理，這是本文證明的1個特點。
關鍵詞：連續函數，閉區間，最大、最小值定理，介值性定理，1致連續性定理，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.
