Math 2.0

counterexample

x^2 sin(1/x) is differentiable with discontinuous derivative

Counterexample

Let f(x)=x2sin⁡(1/x)f(x) = x^2 \sin(1/x) for x≠0x \ne 0, and let f(0)=0f(0) = 0. Then ff is differentiable on R\mathbb{R}, but f′f' is not continuous at 00. At 00, the difference quotient is hsin⁡(1/h)h \sin(1/h), which tends to 00. So f′(0)=0f'(0) = 0. For x≠0x \ne 0, f′(x)=2xsin⁡(1/x)−cos⁡(1/x)f'(x) = 2 x \sin(1/x) - \cos(1/x). This has no limit as x→0x \to 0, because cos⁡(1/x)\cos(1/x) has none.

Remarks

So ff is differentiable but not of class C1C^1. The discontinuity of f′f' at 00 is essential. By Darboux's theorem, a derivative can have no jump discontinuities.

Explanations

L1 human

The factor x2x^2 flattens the oscillation of sin⁡(1/x)\sin(1/x) enough to give a slope 00 at the origin. But near the origin the slopes keep swinging between about −1-1 and 11.

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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 functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionFermat'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 …statementDerivative function, definition, no evidence levelDerivative functiondefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementHigher derivative, definition, no evidence levelHigher derivativedefinitionBoundedness theorem, statement, no evidence levelBoundedness theoremstatementContinuously differentiable function, definition, no evidence levelContinuously differen…definitionExtreme value theorem, statement, no evidence levelExtreme value theoremstatementDarboux's theorem, statement, no evidence levelDarboux's theoremstatementx^2 sin(1/x) is differentiable with discontinuous derivative, counterexample, no evidence levelx^2 sin(1/x) is diffe…counterexample
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. Differentiable function definition
  16. Fermat's theorem on stationary points statement
  17. One-sided derivative definition
  18. Convergent sequence definition
  19. Every sequence has a monotone subsequence statement
  20. Monotone convergence theorem for sequences statement
  21. Derivative function definition
  22. Bolzano–Weierstrass theorem statement
  23. Higher derivative definition
  24. Boundedness theorem statement
  25. Continuously differentiable function definition
  26. Extreme value theorem statement
  27. Darboux's theorem statement
  28. x^2 sin(1/x) is differentiable with discontinuous derivative counterexample

Sources

  1. Web https://en.wikipedia.org/wiki/Differentiable_function, Differentiability classes

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