Fun with Calculus 1: a dynamical re-analysis of fundamental theorems in calculus

Daniel Reem

Published 2007-09-27, updated 2016-06-30Version 5

Although a first course in calculus is not always a pleasant experience for fresh students, a re-examination of its material using the more sophisticated tools and ways of thinking obtained in later stages can be a real fun for experts, advanced students and many others. Here we try to achieve something in this direction by playing with, and looking for new horizons in, three fundamental theorems in calculus and related issues. We start the trek by going to the extreme, namely discussing the extreme value theorem regarding the extreme (optimal) values of a continuous function defined on a compact space. Two short proofs of this theorem are presented: "the programmer proof" that suggests a method (which is practical in down-to-earth settings) to approximate, to any required precision, the extreme values in a metric space setting, and an abstract space proof ("the level-set proof") for semicontinuous functions defined on compact topological spaces. In the intermediate part we discuss the intermediate value theorem which is generalized to a class of discontinuous functions, the meaning of the intermediate value property is re-examined, and a fixed point theorem for (very) discontinuous functions is established. The trilogy reached the final frontier in a discussion on uniform continuity in which the optimal delta of the given epsilon is obtained explicitly, several properties of it are derived, and necessary or sufficient conditions for the uniform continuity of a function are established, among them the classical one in which the space is compact and the function is continuous. Have fun!