Exercise 9.6

Question

Exercise 6: 
    If $f(0,0)=0$ and
    $$
        f(x,y)=\frac{xy}{x^2+y^2}\qquad\text{if }(x,y)\ne(0,0),
    $$
    prove that $(D_1f)(x,y)$ and $(D_2f)(x,y)$ exist at every point of $\R2$, although $f$ is not continuous at $(0,0)$.

analambanomenos · Accepted Answer

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The usual rules of differentiaion show that the partial derivatives of $\R2$ at $(x,y)\ne(0,0)$ are
    $$
        D_1f(x,y)=\frac{y(y^2-x^2)}{(x^2+y^2)^2} \qquad D_2f(x,y)=\frac{x(x^2-y^2)}{(x^2+y^2)^2}
    $$
    And since $f$ is equal to 0 everywhere along the $x$ and $y$ axes, they also exist and are equal to 0 at $(0,0)$.

For $x\ne 0$, $f(x,x)=x^2/(x^2+x^2)=1/2$. Since $f(0,0)=0$, $f$ is not continuous along the line $y=x$ at $(0,0)$.

Exercise 9.6

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Exercise 9.6

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