IGNOU BMTC 102 2026 - Check the continuity and differentiability of the function at (0,0) wh

Question

Check the continuity and differentiability of the function at (0,0) where

$$f(x, y) = \begin{cases} \frac{2x^3 y}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & \text{otherwise} \end{cases}$

09 Jan 2026

Answer :

Word Count : 291

To check the continuity of ( f(x, y) ) at ((0,0)), consider the limit [ \lim_{(x,y) \to (0,0)} \frac{2x^3 y}{x^2 + y^2}. ] Using polar coordinates, set ( x = r \cos \theta ), ( y = r \sin \theta ). Then [ f(x, y) = \frac{2 (r \cos \theta)^3 (r \sin \theta)}{(r \cos \theta)^2 + (r \sin \theta)^2} = \frac{2 r^4 \cos^3 \theta \sin \theta}{r^2} = 2 r^2 \cos^3 \theta \sin \theta. ] As ( r \to 0 ), ( 2 r^2 \cos^3 \theta \sin \theta \to 0 ), independent of ( \theta ). Therefore, [ \lim_{(x,y) \to (0,0)} f(x, y) = 0 = f(0,0), ] so ( f )

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