Math 2.0

statement theorem

Darboux's theorem

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.

Explanations

L1 human

The derivative of a function cannot skip values, just as a continuous function cannot. For example, no function has derivative 00 on the left of a point and 11 on the right.

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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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

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Sources

  1. Web https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis), Statement of the theorem

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