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ISI2015-MMA-71

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Let $$f(x,y) = \begin{cases} 1, & \text{ if } xy=0, \\ xy, & \text{ if } xy \neq 0. \end{cases}$$ Then

  1. $f$ is continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ exists
  2. $f$ is not continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ exists
  3. $f$ is continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ does not exist
  4. $f$ is not continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ does not exist
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Answer: B

Approaching (0,0) from either x=y gives, f(0,0)=0<1. Hence, not continuous. But the partial differential exists at (0,0).

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