Word Count : 372

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We are to check the continuity and differentiability at $(0,0)$ for $$ f(x,y) = \begin{cases} \dfrac{2x^3y}{x^2+y^2}, & (x,y)\neq(0,0)\\[4pt] 0, & (x,y)=(0,0) \end{cases} $$ --- ### 1. Check Continuity at (0,0) A function $f$ is continuous at $(0,0)$ if $$ \lim_{(x,y)\to(0,0)} f(x,y) = f(0,0) = 0 $$ So we check the limit: $$ \lim_{(x,y)\to(0,0)} \frac{2x^3y}{x^2+y^2} $$ Using Polar Coordinates: Let $$ x = r \cos\theta,\quad y = r \sin\theta $$ Then $$ f(r\cos\theta,r\sin\theta) = \frac{2(r\cos\theta)^3 _________ _______ _____ ________ ___ _________ __________ _______ ______ ____ ____.
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