Authors: Steven Kenneth Kauffmann
This tutorial explores the relation of the local concept of a function's continuity to its global consequences on closed intervals, such as a continuous function's unavoidable boundedness on a closed interval, its attainment of its least upper and greatest lower bounds on that interval, and its unavoidable assumption on that closed interval of all of the values which lie between that minimum and maximum. In a nutshell, continuous functions map closed intervals into closed intervals. It is understandable that verifying this local-to-global fact involves subtle and very intricate manipulation of the least-upper-bound/greatest-lower-bound postulate for the real numbers. In conjunction with the basic inequality properties of integrals, this continuous-function fact immediately implies the integral form of the mean-value theorem, which is parlayed into its differential form by the fundamental theorem of the calculus. Taylor expansion and its error estimation are further developments which are intertwined with these fascinating concepts.
Comments: 4 Pages.
Unique-IP document downloads: 9 times
Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.
Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.