AWhat changed
A derivative measures slope. It is tempting to assume that slopes must change continuously, but this is false. Darboux proves a different constraint: if a finite derivative exists at every point of an interval, it takes every value between any two of its values. The proof subtracts a line with an intermediate slope. The resulting function has an interior extremum, where its derivative vanishes, so the original derivative takes the desired slope. The same paper gives examples showing why this intermediate-value property can survive even when the derivative is discontinuous.
Scientific scope
A derivative measures slope. It is tempting to assume that slopes must change continuously, but this is false. Darboux proves a different constraint: if a finite derivative exists at every point of an interval, it takes every value between any two of its values. The proof subtracts a line with an intermediate slope. The resulting function has an interior extremum, where its derivative vanishes, so the original derivative takes the desired slope. The same paper gives examples showing why this intermediate-value property can survive even when the derivative is discontinuous. The derivative must exist finitely throughout the interval, not merely almost everywhere. The intermediate-value property does not imply continuity, and a function with that property need not be a derivative. The subsequent primitive-function criterion additionally assumes integrability. This record covers the derivative theorem, not ownership of all discontinuous-function examples, Riemann integration or the mean-value theorem. The manuscript is signed in1874; the journal volume was published in1875.
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 derivative must exist finitely throughout the interval, not merely almost everywhere. The intermediate-value property does not imply continuity, and a function with that property need not be a derivative. The subsequent primitive-function criterion additionally assumes integrability. This record covers the derivative theorem, not ownership of all discontinuous-function examples, Riemann integration or the mean-value theorem. The manuscript is signed in1874; the journal volume was published in1875.
Contributors
- The named authors in the research record · SCIENCE: Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred.
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