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If $$f(0,0)=0$$ and
$f(x, y)=\frac{x y}{x^{2}+y^{2}} \quad \text { if }(x, y) \neq(0,0)$
prove that $$\left(D_{1} f\right)(x, y)$$ and $$\left(D_{2} f\right)(x, y)$$ exist at every point of $$R^{2}$$, although $$f$$ is not continuous at $$(0,0)$$.
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At any point $$(x, y)$$ except $$(0,0)$$ the differentiability of $$f(x, y)$$ follows from the rules for differentiation and the principles of Chapter 5 . At $$(0,0)$$ it is a routine computation to verify that both partial derivatives equal zero:
$\left(D_{1} f\right)(0,0)=\lim _{h \rightarrow 0} \frac{f(h, 0)-f(0,0)}{h}=0 .$
However, $$f(x, y)$$ is not continuous at $$\left(0,0\right.$$, since $$f(x, x)=\frac{1}{2}$$ for all $$x \neq 0$$, and hence $$\lim _{x \rightarrow 0} f(x, x)=\frac{1}{2} \neq f(0,0)$$.
by Diamond (55.6k points)

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