Denjoy–Luzin theorem

Every derivative is approximately continuous, so its set of discontinuities has Lebesgue measure zero. Approximate continuity tests values using density within shrinking neighborhoods rather than every nearby point.

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Approximate continuity: A function is approximately continuous at a point when its nearby values converge in the sense of density, allowing sparse exceptions. This is the precise regularity property that the theorem guarantees for every derivative.
Lebesgue density theorem: Almost every point of a measurable set is a density point of that set. Density points turn measure-theoretic control of value sets into approximate continuity.
Arnaud Denjoy: Arnaud Denjoy was a French mathematician whose work included real analysis and the theory of derivatives. Denjoy’s investigations of derivatives are part of the theorem’s historical lineage.
Volterra's function: A differentiable function whose derivative is discontinuous at every point of a dense countable set. It illustrates that a derivative may have many discontinuities while still satisfying the theorem.
Derivative: The derivative of a function at a point is the limit of its difference quotients, when that limit exists. The theorem applies to derivatives, whether or not those derivatives are continuous.
Lusin's theorem: A measurable function on a finite-measure set is continuous on a compact subset whose complement has arbitrarily small measure. It provides a nearby regularity principle, though it does not itself establish approximate continuity of derivatives.
Nikolai Luzin: Nikolai Luzin was a Russian mathematician known for work in measure theory, descriptive set theory, and analysis. Luzin’s name is attached to the theorem’s measure-theoretic regularity result.
Thomae's function: A function equal to zero at irrational numbers and to the reciprocal of the denominator at reduced rational numbers. Its dense discontinuities show why ordinary density and measure distinctions matter when studying irregular functions.
Lebesgue measure: Lebesgue measure assigns lengths, areas, or volumes to suitable subsets of Euclidean space. The theorem measures the exceptional discontinuity set using Lebesgue measure.
Darboux's theorem: Every derivative has the intermediate value property, even when it is discontinuous. Derivative values cannot jump across gaps, a structural constraint alongside approximate continuity.