David Álvarez Rosa builds the Fundamental Theorem of Calculus from scratch: Riemann sums, Fermat, Rolle, MVT, then the FTC proof with full epsilon rigor.
Key Takeaways
Riemann integrability is defined via the epsilon-partition criterion: for every epsilon > 0, a partition exists where upper minus lower Darboux sum is less than epsilon.
Boundedness alone (not continuity) is required for subinterval infima and suprema to be finite; Lebesgue’s criterion fully characterizes Riemann integrability.
The proof chain is layered: Fermat’s Proposition establishes zero derivative at extrema, Rolle’s Theorem builds on that, MVT builds on Rolle.
FTC proof applies MVT to F on each subinterval, producing a sample point z_k that sandwiches F(b) - F(a) between lower and upper sums.
The practical payoff: computing the area under a curve reduces to evaluating an antiderivative at two endpoints; no partition arithmetic needed.
Hacker News Comment Review
Comment activity is minimal and low-signal at the time of indexing; no technical disputes or implementation caveats surfaced in the thread.
One commenter linked the Wikipedia FTC article without elaboration, suggesting the thread attracted readers cross-referencing rather than critiquing the proof.
A question about the site’s font was the only other comment, indicating the post’s visual presentation drew as much attention as its mathematical content.