# Rigorous Calculus: ε-δ Analysis

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

$\frac{f(x+i)-f(x)}{i}$  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 < $\frac{f(x+i)-f(x)}{i}$ < 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...”