Mathematics – Classical Analysis and ODEs
Scientific paper
2007-09-27
Mathematics
Classical Analysis and ODEs
8 pages; Several changes in the title, abstract, and the introduction section
Scientific paper
New proofs and improvements of three classical theorems in analysis are presented. Although these theorems are well-known, and have been extensively investigated over the years, it seems that new light can be shed on them. We first present a quantitative necessarily and sufficient condition for a function to be uniformly continuous, and as a by-product we obtain explicitly the optimal delta for the given epsilon. The uniform continuity of a continuous function defined on a compact metric space follows as a simple consequence. We proceed with the extreme value theorem and present a ``programmer's proof'', a proof which does not use the costume argument of proving boundedness first. We finish with the intermediate value theorem, which is generalized to a class of discontinuous functions, and, in addition, the meaning of the intermediate value property is re-examined and a fixed point theorem for (very) discontinuous functions is established. At the end we discuss briefly in which sense the proofs are constructive.
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