Rigorous Analysis epsilon-delta (ε-δ)

Cauchy gave **epsilon-delta** the rigor to Analysis, Weierstrass ‘* arithmatized*‘ it to become the standard language of modern analysis.

1) Limit was first defined by Cauchy in “** Analyse Algébrique**” (1821)

2) Cauchy repeatedly used ‘Limit’ in the book Chapter 3 “** Résumé des Leçons sur le Calcul infinitésimal**” (1823) for ‘derivative’ of f as the limit of

when i -> 0

3) He introduced ε-δ in Chapter 7 to prove ‘**Mean Value Theorem**‘: Denote by (ε , δ) 2 small numbers, such that 0< i ≤ δ , and for all x between (x+i) and x,

f ‘(x)- ε < < f'(x)+ ε

4) These ε-δ Cauchy’s proof method became the standard definition of **Limit of Function** in Analysis.

5) They are notorious for causing widespread discomfort among future math students. In fact, when it was first introduced by Cauchy in the **Ecole Polytechnique** Lecture, the French Napoleon top students booed at him and Cauchy received **warning** from the school.

Note 1: From the textbook ‘**Calculus**‘ (1980, USA):

“*If can’t understand the ‘ε-δ’ definition, just memorize it like a poem – at least better than saying it wrongly.*”

E.g. “**For all ε>0, there exists δ>0, …**”

Note 2: George Polya: “*The students are not trained in ‘ε-δ’, teaching them Calculus is like dropping these rules from the sky.*..”

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