Re:Calculus with Julia
by dan-robertson on Monday May 20, 2024 @11:18AM
(#40416492)
Ah thanks. My understanding was the whole epsilon-delta stuff one typically learns in a first course in analysis was due to Cauchy.
No, epsilon-delta in the modern form was invented by Karl Weierstrass in 1861.
Cauchy's approach was to define an infinitesmal as a sequence approaching 0, and then to use the old infinitesmal definition of a derivative. This almost works. One of the places where it falls apart is defining the chain rule when the inner derivative is 0, because the sequence may wind up with 0/0 infinitely often.
But it is from this definition that we get Cauchy sequences today.
But I have to correct you on the history. Essentially none of the material in a modern analysis course is the same as in Cauchy's day. You identify integration as coming after. But Cauchy didn't have a definition of a limit, axioms for the real numbers, a construction of the real numbers, set theory, the modern idea of open and closed sets, and so on and so forth.