Math 2.0

Real functions Differentiation Theorem

Darboux's theorem

Builds on Extreme value theoremFermat's theorem on stationary pointsOne-sided derivative

Statement

Let a<ba < b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be differentiable on [a,b][a, b], with one-sided derivatives at the endpoints. Let yy be a real number strictly between f′(a)f'(a) and f′(b)f'(b). Then there is c∈(a,b)c \in (a, b) with f′(c)=yf'(c) = y.

Remarks

Assume f′(a)<y<f′(b)f'(a) < y < f'(b). The function g(x)=f(x)−yxg(x) = f(x) - y x has g′(a)<0<g′(b)g'(a) < 0 < g'(b). So its minimum on [a,b][a, b], given by the extreme value theorem, is at an interior point cc. Fermat's theorem gives g′(c)=0g'(c) = 0. So a derivative has the intermediate value property, even when it is not continuous. In particular, a derivative has no jump discontinuities. Mathlib states it as exists_hasDerivWithinAt_eq_of_gt_of_lt.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementOne-sided derivative, definition, no evidence levelOne-sided derivativedefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementDarboux's theorem, statement, no evidence levelDarboux's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. One-sided limit definition
  11. Bounded sequence definition
  12. Limit of a sequence definition
  13. Monotone sequence definition
  14. Continuous function definition
  15. Fermat's theorem on stationary points statement
  16. One-sided derivative definition
  17. Convergent sequence definition
  18. Every sequence has a monotone subsequence statement
  19. Monotone convergence theorem for sequences statement
  20. Bolzano–Weierstrass theorem statement
  21. Boundedness theorem statement
  22. Extreme value theorem statement
  23. Darboux's theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis), Statement of the theorem
About this page
Tag0Q5A47J48D
URIm2://0Q5A47J48D
Aliasesintermediate-value-theorem-for-derivatives
Kindstatement, theorem
WritingW1 stated
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0Q5A47J48D.
UsesExtreme value theorem, Fermat's theorem on stationary points, One-sided derivative
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