}(a, b) \text { iff } \lim _{(x, y) \rightarrow(a, b)} f(x, y)=f(a, b) \begin{aligned} &\text { Note } 2 \text { If }(x, y)=(0,0), \text { by definition } f_{x}(a, b)=\lim _{h \rightarrow 0} \frac{f(a+h, b)-f(a, b)}{h}\\ &\text { we have } f_{x}(0,0) \text { and } f_{y}(0,0)=\lim _{h \rightarrow 0} \frac{f(0+h, 0)-f(0,0)}{h} \text { . } \end{aligned} \text { Note3 If }(x, y) \neq(0,0), \quad f_{x}(x, y)=\frac{\partial}{\partial x}\left(\frac{x y}{x^{2}+y^{2}}\right), \quad f_{y}(x, y)=\frac{\partial}{\partial y}\left(\frac{x y}{x^{2}+y^{2}}\right) (a) Show that f(x,y) is not continuous at (0,0). (b) Show that f.(0,0) and fy(0,0) exist. (c) Show that fr and fy are not differentiable at (0,0).

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