SWhat changed
Rademacher proves that a finite-dimensional Lipschitz function is totally differentiable outside a Lebesgue-null set. His original stronger sufficient condition uses a finite integrable upper local slope, and the paper connects almost-everywhere linearization to Jacobians and a change-of-variables formula for one-to-one transformations under its stated hypotheses.
Scientific scope
Rademacher proves that a finite-dimensional Lipschitz function is totally differentiable outside a Lebesgue-null set. His original stronger sufficient condition uses a finite integrable upper local slope, and the paper connects almost-everywhere linearization to Jacobians and a change-of-variables formula for one-to-one transformations under its stated hypotheses. The displayed proof is written for two variables, with extension to any finite number stated explicitly; finite-dimensional vector outputs follow componentwise, not arbitrary infinite-dimensional Banach targets. The derivative is a total linear approximation almost everywhere, not everywhere and not merely the separate existence of partial derivatives. The upper-local-slope variant requires its original finiteness/integrability conditions. The associated integral transformation is one-to-one and satisfies the original regularity/measurability assumptions, not a general noninjective multiplicity formula.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The displayed proof is written for two variables, with extension to any finite number stated explicitly; finite-dimensional vector outputs follow componentwise, not arbitrary infinite-dimensional Banach targets. The derivative is a total linear approximation almost everywhere, not everywhere and not merely the separate existence of partial derivatives. The upper-local-slope variant requires its original finiteness/integrability conditions. The associated integral transformation is one-to-one and satisfies the original regularity/measurability assumptions, not a general noninjective multiplicity formula.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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