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For (x, y) ∈ R2, let
  • a)
     fx and fexists at (0, 0 ) and f is continuous at (0, 0)
  • b)
    fx and fexists at (0,0) and f is discontinuous at (0,0)
  • c)
    fx and fy does not exist at (0, 0) and f is continuous at (0, 0)
  • d)
    fx and fdoes not exists at (0 , 0) and f is discontinuous at (0, 0)
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
For (x, y) ∈ R2, leta)fx and fyexists at (0, 0 ) and fis continuo...
Let us approaches (0, 0) along the line y = mx which passes through origin. Put y = mx, we get

which depends on m. Hence,  does not exists, Therefore f(x, y) is discontinuous at (0,0).
Now, 
and 

Hence, f(x,y) is discontinuous at (0, 0). But both the partial derivative fx and fy exists at origin.
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Most Upvoted Answer
For (x, y) ∈ R2, leta)fx and fyexists at (0, 0 ) and fis continuo...
Let us approaches (0, 0) along the line y = mx which passes through origin. Put y = mx, we get

which depends on m. Hence,  does not exists, Therefore f(x, y) is discontinuous at (0,0).
Now, 
and 

Hence, f(x,y) is discontinuous at (0, 0). But both the partial derivative fx and fy exists at origin.
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For (x, y) ∈ R2, leta)fx and fyexists at (0, 0 ) and fis continuous at (0, 0)b)fx and fyexists at (0,0) and f is discontinuous at (0,0)c)fx and fydoes not exist at (0, 0) and fis continuous at (0, 0)d)fxand fydoes not exists at (0 , 0) and fis discontinuous at (0, 0)Correct answer is option 'B'. Can you explain this answer?
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