Calculus I · Unit 2A · exploration
Darboux's Theorem: Derivatives Cannot Jump
An Intermediate Value Property Without Continuity
A derivative may fail to be continuous, but it cannot have a jump discontinuity. Darboux's Theorem states that if takes values and at two points, then it takes every value between and somewhere between those points.
The result is surprising because the conclusion resembles the Intermediate Value Theorem, yet no continuity of is assumed.
Read this graph as text
A derivative may be rough, but it cannot jump over slope values. The left panel shows a step-shaped candidate with a genuine jump; Darboux's Theorem rules it out as the derivative of any everywhere-differentiable function on that interval. The right panel crosses every intermediate height. The left graph goes from slope -1 to slope 1 without ever taking slope 0 . A derivative cannot do that. It may oscillate or fail to be continuous in subtler ways, but whenever it takes two slope values it must take every value between them somewhere in between.
The visual uses labeled positions, solid and dashed line styles, and written descriptions so a derivative may be rough, but it cannot jump over slope values does not depend on color.
Why it matters: The side-by-side comparison should prevent the false inference that "derivatives need not be continuous" means "any discontinuous graph can be a derivative." The visual introduces a global restriction on local slope data and gives students a quick way to reject an impossible proposed derivative graph.
The left panel shows a step-shaped candidate with a genuine jump; Darboux's Theorem rules it out as the derivative of any everywhere-differentiable function on that interval. The right panel crosses every intermediate height.
The proof uses the Extreme Value Theorem and Fermat's theorem on an auxiliary function. To show that attains a value between and , consider
Its derivatives at the endpoints have opposite signs. An interior extremum of then produces , which means .
A boundary on possible derivative graphs
A graph proposed as may be discontinuous, but a genuine jump rules it out immediately. This is one of the first examples where a theorem imposes a hidden global restriction on local slope data.
After the explanation
Use the section idea
Use these optional explorations to see the deeper analysis behind familiar derivative rules.
Differentiability is a local linear approximation property with consequences beyond computation.
Return here after the core path is secure, and connect each abstraction to a concrete derivative example.
Collecting formal language without linking it to the local linear model it describes.
Can you restate the advanced claim in ordinary language and test it on an example?
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