Word Count : 427

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To show that the function \( f(x, y) \) is not continuous at \( (0, 0) \), we need to demonstrate that the limit of \( f(x, y) \) as \( (x, y) \) approaches \( (0, 0) \) does not exist or does not equal \( f(0, 0) \). The function is defined as: \[ f(x, y) = \begin{cases} y \sin{\frac{1}{x}} + x \sin{\frac{1}{y}}, & \text{if } x \ne 0 \text{ and } y \ne 0 \\ 1, & \text{otherwise} \end{cases} \] At \( (0, 0) \), \( f(0, 0) = 1 \). ### Step 1: Approach along the x-axis (\( y = 0 \)) ___ ________ _____ ______ _____ _______ _____ _______ _____.
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