English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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, canbe a real fun for experts, advanced students, etc. We start by going to theextreme, namely we present two proofs of the Extreme Value Theorem: "theprogrammer proof" that suggests a method (which is practical in down-to-earthsettings) to approximate, to any required precision, the extreme values of thegiven function in a metric space setting, and an abstract space proof ("thelevel-set proof") for semicontinuous functions defined on compact topologicalspaces. Next, in the intermediate part, we consider the Intermediate ValueTheorem, generalize it to a wide class of discontinuous functions, andre-examine the meaning of the intermediate value property. The trek reaches thefinal frontier when we discuss the Uniform Continuity Theorem, generalize it,re-examine the meaning of uniform continuity, and find the optimal delta of thegiven epsilon. Have fun!

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