The argument connects a local condition on discontinuities with the upper and lower sums of Riemann integration. Compactness supplies the finite control needed to finish the proof.
Background
Riemann integration, compactness, and sets of measure zero.
Read the full exposition
The complete argument is available as a PDF.
Revision history · ;
Revised September 22, 2026: included the zero-length interval, corrected two strict oscillation bounds to non-strict bounds, and supplied a direct Darboux proof of the composition corollary that uses neither direction of Lebesgue’s criterion. Revised October 7, 2026: rebuilt the editable source, clarified overlapping covers and closed-cell estimates, and developed the Thomae and Cantor-set examples. The PDF and TeX source agree.