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Let . then
  • a)
    f is continuous at (0, 0) and the partial derivatives fx, fy exist at every point of ℝ2
  • b)
    f is discontinuous at (0,0) and fx, fy exist at every point of ℝ2
  • c)
    f is discontinuous at (0,0) and fx, fy exist only at (0,0)
  • d)
    None of the above
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Let. thena)f is continuous at (0, 0) and the partial derivatives fx, f...
For continuity at (0,0)
Let x = r cos θ, y = r sinθ
If (x,y)→ (0,0) then r→0
= cosθ.sinθ, which depends on the choice of θ
hence limit do not exist at (0,0). Therefore f(x, y) is discontinuous at (0,0).
Partial derivatives at (0,0)
∴ both partial derivatives exist at (0,0).
Also at any other point
Hence partial derivatives exist at every point of ℝ2.
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Let. thena)f is continuous at (0, 0) and the partial derivatives fx, fy exist at every point of2b)f is discontinuous at (0,0) and fx, fy exist at every point of2c)f is discontinuous at (0,0) and fx, fy exist only at (0,0)d)None of the aboveCorrect answer is option 'A'. Can you explain this answer?
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Let. thena)f is continuous at (0, 0) and the partial derivatives fx, fy exist at every point of2b)f is discontinuous at (0,0) and fx, fy exist at every point of2c)f is discontinuous at (0,0) and fx, fy exist only at (0,0)d)None of the aboveCorrect answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let. thena)f is continuous at (0, 0) and the partial derivatives fx, fy exist at every point of2b)f is discontinuous at (0,0) and fx, fy exist at every point of2c)f is discontinuous at (0,0) and fx, fy exist only at (0,0)d)None of the aboveCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let. thena)f is continuous at (0, 0) and the partial derivatives fx, fy exist at every point of2b)f is discontinuous at (0,0) and fx, fy exist at every point of2c)f is discontinuous at (0,0) and fx, fy exist only at (0,0)d)None of the aboveCorrect answer is option 'A'. Can you explain this answer?.
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