GATE 2024 ST – Question 37

Calculus · Multivariable Calculus · 2 marks · Multiple choice

Define f(x,y)=(x³−y³)/(x²+y²) off the origin and f(0,0)=0. Which is not true?

f continuous at the origin
f_x(0,0)≠f_y(0,0)
f_x continuous at the origin
f_y not continuous at the origin

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Correct answer: (C) f_x continuous at the origin

Explanation

Bound |f|≤2√(x²+y²), proving continuity. Along the axes f(x,0)=x,f(0,y)=−y, so f_x(0,0)=1,f_y(0,0)=−1. Yet for y≠0, f_x(0,y)=0, so f_x is not continuous at zero. Also f_y(x,0)=0 for x≠0, proving its discontinuity. The false statement is **C**.

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