Math 2.0

Real functions Differentiation Counterexample

Smooth non-analytic function

Builds on Smooth functionAnalytic functionTaylor series

Counterexample

Let f(x)=e−1/xf(x) = e^{-1/x} for x>0x > 0 and f(x)=0f(x) = 0 for x≤0x \le 0. Then ff is smooth on ℝ\mathbb{R}, and f(n)(0)=0f^{(n)}(0) = 0 for all nn. So its Taylor series at 00 is the zero series. But f(x)>0f(x) > 0 for x>0x > 0, so ff is not analytic at 00.

Remarks

For x>0x > 0, each derivative has the form pn(1/x)e−1/xp_n(1/x) e^{-1/x} for a polynomial pnp_n. It tends to 00 as x→0+x \to 0^+, because e−1/xe^{-1/x} decays faster than any power of xx. Such functions give smooth bump functions and smooth partitions of unity. Mathlib defines this function as expNegInvGlue.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionSeries, definition, no evidence levelSeriesdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionPower series, definition, no evidence levelPower seriesdefinitionDerivative function, definition, no evidence levelDerivative functiondefinitionAnalytic function, definition, no evidence levelAnalytic functiondefinitionHigher derivative, definition, no evidence levelHigher derivativedefinitionTaylor polynomial, definition, no evidence levelTaylor polynomialdefinitionContinuously differentiable function, definition, no evidence levelContinuously differen…definitionSmooth function, definition, no evidence levelSmooth functiondefinitionTaylor series, definition, no evidence levelTaylor seriesdefinitionSmooth non-analytic function, counterexample, no evidence levelSmooth non-analytic f…counterexample
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  1. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Continuity at a point definition
  6. Derivative definition
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  8. Continuous function definition
  9. Differentiable function definition
  10. Power series definition
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  12. Analytic function definition
  13. Higher derivative definition
  14. Taylor polynomial definition
  15. Continuously differentiable function definition
  16. Smooth function definition
  17. Taylor series definition
  18. Smooth non-analytic function counterexample

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4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Non-analytic_smooth_function, An example function
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Tag09RTYYHXKA
URIm2://09RTYYHXKA
Kindcounterexample
WritingW0 stub
MSC26E10
UsesSmooth function, Analytic function, Taylor series
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