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Consider the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ defined by $$ f(0,0)=0, \quad f(x, y)=\frac{x y}{x^{2}+y^{2}}, \quad(x, y) \neq(0,0) . $$ Prove that the directional derivative of $f$ at $(0,0)$ exists in all directions. Is $f$ continuous at $(0,0)$ ? Justify your answer.
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