The Pillars of Analysis: What Holds Calculus Up

Calculus runs the modern world, yet for a century and a half after Newton nobody could say exactly what it rested on. This video draws the whole of analysis as one building: the ground is the completeness of the real line (no holes), the five pillars are the five ways of taking a limit (limits, continuity, the derivative, the integral, series), and the beams between them are the theorems of a first analysis course. Three beams are built by hand: the epsilon-N game for 1/n, the intermediate value theorem by bisection, and the fundamental theorem of calculus from a thin strip. The episode ends with the classic failure x^n on [0, 1] and the idea that repairs it, uniform convergence. Made for first-year students and curious adults who know derivatives and integrals as rules and want to see what holds them up. The derivative as a zoomed slope (from The Map of Mathematics) is enough background. Chapters 0:00 What holds calculus up? 0:59 The ground: fractions have holes 1:43 Dedekind and the least upper bound 2:29 Pillar 1: limits as a game 3:22 The definition, and the first beam 4:13 Pillar 2: continuity 5:08 Bolzano cuts an interval in half 5:42 The intermediate value theorem 6:37 Pillar 3: the derivative 7:18 Corners, Weierstrass, the mean value theorem 8:21 Pillar 4: the integral as a squeeze 9:19 The fundamental theorem of calculus 10:17 Pillar 5: series 11:19 Cauchy's false theorem and uniform convergence 12:21 The whole building: the answer 13:16 The upper floors 13:45 Next: Lebesgue, and credits Further reading S. Abbott, Understanding Analysis (2nd ed., Springer, 2015) M. Spivak, Calculus (4th ed., Publish or Perish, 2008) D. Bressoud, A Radical Approach to Real Analysis (2nd ed., MAA, 2007) Next on the channel: the Lebesgue integral, why Riemann's rectangles give up on the function that is 1 on fractions and 0 elsewhere. The view from above is the episode The Map of Mathematics. Narration uses a synthetic voice; the script, mathematics and animations are written and checked for this channel.