Making Calculus Rigorous
Newton and Leibniz invented calculus in the 17th century using "infinitely small" quantities that were philosophically incoherent — the Bishop Berkeley famously mocked them as "ghosts of departed quantities." The machinery worked empirically, but the foundations were shaky. Cauchy and Weierstrass in the 19th century rebuilt analysis on the epsilon-delta definition of a limit, eliminating infinitesimals in favor of precise inequalities.
The Epsilon-Delta Definition of a Limit
The limit of f(x) as x approaches a equals L means: for every epsilon greater than 0, there exists delta greater than 0 such that whenever 0 is less than the absolute value of x minus a and that is less than delta, the absolute value of f(x) minus L is le
The Completeness of the Real Numbers
The real numbers are complete: every Cauchy sequence — a sequence where the terms eventually get arbitrarily close to each other — converges to a limit in R. The rational numbers Q are not complete: the sequence 1, 1.4, 1.41, 1.414, 1.4142, ... is Cauchy
Continuity and Uniform Continuity
A function f is continuous at a point a if the limit of f(x) as x approaches a equals f(a) — the limit exists and equals the function value. Uniform continuity is a global version: the same delta works for all points simultaneously, not just at each point
The Intermediate and Mean Value Theorems
The Intermediate Value Theorem: if f is continuous on [a, b] and c is any value between f(a) and f(b), there exists x in [a, b] with f(x) = c. Proof uses completeness — bisect the interval, track which half contains a sign change, take the limit of the mi
Taylor Series and Error Bounds
A smooth function f equals its Taylor series sum over n from 0 to N of the nth derivative of f at a divided by n factorial times (x minus a) to the n, plus a remainder R_N(x). The Lagrange remainder formula bounds R_N by the maximum of the (N+1)th derivat
Go Deeper: Real Analysis and Topology
Real analysis studies the real line — a specific topological space. Topology studies what properties survive continuous deformation, abstracting away particular geometry. It sits beneath analysis, connects to algebra through algebraic topology, and leads to the deepest structures in all of mathematics.
Understand Algebraic Topology and the Foundations of Space
The bottom of the rabbit hole — fundamental groups, homology, and the deepest mathematical structures beneath geometry, analysis, and the spaces that cryptographic algorithms operate in.