Fun with "Analysis I": basic theorems in calculus revisited

TL;DR

This paper revisits fundamental calculus theorems using advanced tools, providing new proofs and insights that enhance understanding for experts and students alike.

Contribution

It introduces novel proofs and generalizations of key theorems like EVT, IVT, and uniform continuity, connecting classical analysis with modern approaches.

Findings

Presented two proofs of the Extreme Value Theorem, including a practical 'programmer proof' and an abstract space proof.

Generalized the Intermediate Value Theorem to broader classes of functions and reinterpreted its meaning.

Extended the Uniform Continuity Theorem, offering a new perspective and optimal delta calculations.

Abstract

This note tries to show that a re-examination of a first course in analysis, using the more sophisticated tools and approaches obtained in later stages, can be a real fun for experts, advanced students, etc. We start by going to the extreme, namely we present two proofs of the Extreme Value Theorem: "the programmer proof" that suggests a method (which is practical in down-to-earth settings) to approximate, to any required precision, the extreme values of the given function in a metric space setting, and an abstract space proof ("the level-set proof") for semicontinuous functions defined on compact topological spaces. Next, in the intermediate part, we consider the Intermediate Value Theorem, generalize it to a wide class of discontinuous functions, and re-examine the meaning of the intermediate value property. The trek reaches the final frontier when we discuss the Uniform Continuity Theorem, generalize it, re-examine the meaning of uniform continuity, and find the optimal delta of the given epsilon. Have fun!