On D=[-1,1]2, define f(x,y)=casesx2-y2(x2+y2)2,&(x,y)(0,0),\\0,&(x,y)=(0,0).cases Which is true?

Both first partial derivatives exist at (0,0).

GATE 2020 MA – Question 12

Real Analysis · Multivariable Analysis · 1 mark · Multiple choice

On $D=[-1,1]^2$, define $$f(x,y)=\begin{cases}\dfrac{x^2-y^2}{(x^2+y^2)^2},&(x,y)\ne(0,0),\\0,&(x,y)=(0,0).\end{cases}$$ Which is true?

f is continuous at (0,0).
Both first partial derivatives exist at (0,0).
$\int_D|f|^{1/2}\,dx\,dy$ is finite.
$\int_D|f|\,dx\,dy$ is finite.

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Correct answer: (C) $\int_D|f|^{1/2}\,dx\,dy$ is finite.

Explanation

In polar coordinates, $f=\cos(2\theta)/r^2$. Its magnitude is unbounded near the origin, and along the x-axis the partial-derivative quotient is $1/h^3$, so A and B fail. For the half-power, the radial factor is $|f|^{1/2}r\,dr=|\cos2\theta|^{1/2}dr$, integrable near zero. For $|f|$ it is proportional to $dr/r$, which diverges on any angular sector where the cosine stays away from zero. Hence C.

GATE 2020 MA – Question 12

Explanation

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