Analysis II | Part IB, 2003

Explain what it means for a function f:R2R1f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{1} to be differentiable at a point (a,b)(a, b). Show that if the partial derivatives f/x\partial f / \partial x and f/y\partial f / \partial y exist in a neighbourhood of (a,b)(a, b) and are continuous at (a,b)(a, b) then ff is differentiable at (a,b)(a, b).


f(x,y)=xyx2+y2((x,y)(0,0))f(x, y)=\frac{x y}{x^{2}+y^{2}} \quad((x, y) \neq(0,0))

and f(0,0)=0f(0,0)=0. Do the partial derivatives of ff exist at (0,0)?(0,0) ? Is ff differentiable at (0,0)?(0,0) ? Justify your answers.

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