Math 2.0

Real functions Counterexample

Function with unequal mixed partial derivatives

Builds on Schwarz's theorem for C^2 functionsHigher partial derivatives

Counterexample

One could guess that the mixed partial derivatives fxyf_{xy} and fyxf_{yx} are equal at each point where both exist. This is false. Let f ⁣:ℝ2→ℝf \colon \mathbb{R}^2 \to \mathbb{R} be given by f(x,y)=xy(x2−y2)x2+y2 for (x,y)≠(0,0),f(0,0)=0.f(x, y) = \frac{x y (x^2 - y^2)}{x^2 + y^2} \text{ for } (x, y) \ne (0, 0), \qquad f(0, 0) = 0. Both mixed second partial derivatives of ff exist at (0,0)(0, 0), but ∂∂y(∂f∂x)(0,0)=−1\dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial x}\right)(0, 0) = -1 and ∂∂x(∂f∂y)(0,0)=1\dfrac{\partial}{\partial x}\left(\dfrac{\partial f}{\partial y}\right)(0, 0) = 1.

Remarks

From the definition, ∂f/∂x(0,y)=lim⁡h→0f(h,y)/h=−y\partial f / \partial x (0, y) = \lim_{h \to 0} f(h, y) / h = -y for each yy, and ∂f/∂y(x,0)=lim⁡k→0f(x,k)/k=x\partial f / \partial y (x, 0) = \lim_{k \to 0} f(x, k) / k = x for each xx. Differentiate these once more at 00. The function is of class C1C^1 on ℝ2\mathbb{R}^2, but its second partial derivatives are not continuous at the origin. So Schwarz's theorem does not apply. This example is due to Peano.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionSymmetry of second derivatives, statement, no evidence levelSymmetry of second de…statementC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementSchwarz's theorem for C^2 functions, statement, no evidence levelSchwarz's theorem for…statementFunction with unequal mixed partial derivatives, counterexample, no evidence levelFunction with unequal…counterexample
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  1. Euclidean space object
  2. Function concept
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  4. Function of several variables concept
  5. Limit of a function of several variables concept
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  7. Partial derivative definition
  8. Differentiable on an open set definition
  9. Higher partial derivatives definition
  10. Iterated derivative definition
  11. Symmetry of second derivatives statement
  12. C^k function characterised by continuous derivatives statement
  13. Schwarz's theorem for C^2 functions statement
  14. Function with unequal mixed partial derivatives counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives, Requirement of continuity
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UsesSchwarz's theorem for C^2 functions, Higher partial derivatives

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