RefacteThe mechanism
Turn local oscillation into finite control
The proof groups points by the amount a bounded function can oscillate near them. Covering the exceptional set and using compactness elsewhere turns local information into a finite partition with a small Darboux gap.
What becomes elementary: A pointwise continuity condition becomes a quantitative estimate on upper and lower sums.
Follow the passage from oscillation to a finite cover and then to the partition estimate. Compare the same local-to-global operation in the product proof.
For another construction, explore the mechanism calculi. Edition dates, sources and reference chapters identify the scope of each study.
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: clarified overlapping covers and closed-cell estimates, and developed the Thomae and Cantor-set examples.