Nearness and continuity
Degree GrantorUniversity of Canterbury
Degree NameBachelor of Science with Honours
Nearness relations provide an approach to continuity and limits that makes a clear transition from motivation and intuitive understanding to rigorous analysis. This approach is advocated, in [?], as a means of introducing students to real analysis without excessive abstraction. In the following work, we present the axioms for a nearness space, prove that the natural definition on Rn does give a nearness relation, and then use nearness to define and study the notion of a continuous function. We also prove the equivalence of the nearness definition of continuity and the standard ε-δ definition. Finally, we deal with two pillars of elementary real analysis: the Intermediate Value Theorem and the Extreme Value Theorem.
SubjectsField of Research::01 - Mathematical Sciences
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