Math 2.0

Real functions Counterexample

Partial derivatives do not imply differentiability

Builds on Partial derivativeDifferentiable function of several variablesSeparately continuous but not continuous function

Counterexample

One could guess that a function of several variables is differentiable at a point when all its partial derivatives exist there. This is false. Let f ⁣:ℝ2→ℝf \colon \mathbb{R}^2 \to \mathbb{R} be given by f(x,y)=xyx2+y2f(x, y) = \dfrac{xy}{x^2 + y^2} for (x,y)≠(0,0)(x, y) \ne (0, 0) and f(0,0)=0f(0, 0) = 0. Both partial derivatives of ff exist at (0,0)(0, 0), and both are 00. But ff is not continuous at (0,0)(0, 0), so ff is not differentiable at (0,0)(0, 0).

Remarks

The function is 00 on both axes, so f(h,0)=f(0,h)=0f(h, 0) = f(0, h) = 0 for each hh, and both difference quotients at the origin are 00. On the line y=xy = x the value is 1/21/2, so ff is not continuous at the origin, as in the separately continuous function. A function that is differentiable at a point is continuous there. Even all directional derivatives are not enough. For g(x,y)=x2y/(x4+y2)g(x, y) = x^2 y / (x^4 + y^2) off the origin and g(0,0)=0g(0, 0) = 0, each directional derivative at the origin exists, but g=1/2g = 1/2 on the parabola y=x2y = x^2 minus the origin. If the partial derivatives exist near the point and are continuous at it, then the function is differentiable, by the criterion with continuous partial derivatives.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptParametrised curve, concept, no evidence levelParametrised curveconceptPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity in R^n, concept, no evidence levelContinuity in R^nconceptLimit along different paths, concept, no evidence levelLimit along different…conceptSeparately continuous but not continuous function, counterexample, no evidence levelSeparately continuous…counterexamplePartial derivatives do not imply differentiability, counterexample, no evidence levelPartial derivatives d…counterexample
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.
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  1. Euclidean space object
  2. Function concept
  3. Total derivative concept
  4. Function of several variables concept
  5. Vector-valued function concept
  6. Differentiable function of several variables concept
  7. Limit of a function of several variables concept
  8. Parametrised curve concept
  9. Partial derivative concept
  10. Continuity in R^n concept
  11. Limit along different paths concept
  12. Separately continuous but not continuous function counterexample
  13. Partial derivatives do not imply differentiability counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Partial_derivative, Definition
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URIm2://0TNZC55X3J
Kindcounterexample
WritingW0 stub
MSC26B05
UsesPartial derivative, Differentiable function of several variables, Separately continuous but not continuous function

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