Word Count : 587

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The given function is defined as: \[ f(x, y) = \left\{ \begin{matrix} \frac{2x^3y}{x^2 + y^2}, & \text{if } (x, y) \neq (0, 0) \\ 0, & \text{if } (x, y) = (0, 0) \end{matrix} \right. \] We need to check the continuity and differentiability of the function at the point \((0, 0)\). ### 1. Checking Continuity at \((0, 0)\): A function is continuous at a point if: \[ \lim_{(x,y) \to (0,0)} f(x, y) = f(0, 0) \] We know that \( f(0, 0) = 0 \). Now we need to check if the limit of \( f(x, y) \) as \( (x, y) \to (0, 0) \) is also \( 0 \). #### Approach along different paths: - Along the x-axis (\( y = 0 \)): \[ f(x, 0) = \frac{2x^3(0)}{x^2 + 0^2} = 0 \] So, as \( x \to 0 \), \( f(x, 0) ________ _________ ____ ____ ____ ____.
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