Can a slope skip a value?
For a smooth-looking curve, it seems natural that slopes change continuously. But differentiability of a function does not force continuity of its derivative. Darboux identifies a different universal constraint: when the finite derivative exists everywhere in an interval, it cannot skip an intermediate value. If its slope is one at one location and three at another, the slope must be two somewhere between them. This remains true even when the derivative is discontinuous.
Subtract the slope you want to find
Choose a desired intermediate slope m and subtract the straight line m times x from the original function. At the two selected endpoints, the new derivative has opposite signs. In the appropriate orientation the new function begins by increasing and ends by decreasing, so its maximum occurs at an interior point. At that maximum the derivative is zero. Undoing the subtraction shows that the original derivative equals m there. The argument turns a slope question into an extremum question.
Oscillation differs from a jump
Define a classroom function as x squared times sin of one divided by x away from zero, and set its value at zero to zero. Its derivative at zero is zero, while away from zero the derivative contains a cosine term that keeps oscillating. The derivative is discontinuous at zero, yet it does not create a clean jump that skips all intermediate values. The example shows why discontinuous is a broader category than having a simple step discontinuity.
Do not reverse the implication
Every derivative in the stated scope has the intermediate-value property. It does not follow that every function with that property is a derivative. Nor may the hypothesis be weakened silently to a derivative existing only almost everywhere. A step function illustrates the importance of the condition: it can jump from one level to another without taking intermediate levels, so it cannot be an everywhere-existing finite derivative on that interval. The theorem provides a necessary condition, not a complete characterization.
An attainable source-and-example check
Ask your agent to differentiate the oscillating example away from zero and compute the derivative at zero from its definition. It should explain why a graph sampled at finitely many points cannot establish continuity or the intermediate-value property by itself. Then have it identify the exact derivative hypothesis in the source. The manuscript signature is from1874and the journal publication from1875; those dates serve different roles. A concise checked explanation is a useful contribution without claiming a new theorem.
Scientific scope
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.
Read the original work
Sources and reading locations for this explanation
- Mémoire sur les fonctions discontinues — original 1875 journal scan, especially §IX pp.109–112
Original SectionIXpp109–112,oscillating derivative example and subtract-a-line proof;title/history introductory pages.
This English explanation is an editorial reading view. It does not replace the complete original work or certify its science.
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People and systems
Gaston Darboux
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.
Research locations and affiliations
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Documented countries / regions: France
Original author affiliation is the Ecole Normale in the French memoir. Present-day place labels do not assert nationality,modern-state ownership or historical territorial jurisdiction.