2007/09/27 by Reem, Daniel
#03F99 #26A03 #26A15 #54D05 #90C26 #90C59 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #G.1.0 #G.1.2 #G.1.5 #G.1.6 #H.1.1 #History and Overview (math.HO) #J.2
paper · doi:10.48550/arxiv.0709.4492
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!