Analysis II | Part IB, 2005

Let f:R2Rf: \mathbf{R}^{2} \rightarrow \mathbf{R} be a function. What does it mean to say that ff is differentiable at a point (a,b)(a, b) in R2\mathbf{R}^{2} ? Show that if ff is differentiable at (a,b)(a, b), then ff is continuous at (a,b)(a, b).

For each of the following functions, determine whether or not it is differentiable at (0,0)(0,0). Justify your answers.


f(x,y)={x2y2(x2+y2)1 if (x,y)(0,0)0 if (x,y)=(0,0)f(x, y)= \begin{cases}x^{2} y^{2}\left(x^{2}+y^{2}\right)^{-1} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}


f(x,y)={x2(x2+y2)1 if (x,y)(0,0)0 if (x,y)=(0,0)f(x, y)= \begin{cases}x^{2}\left(x^{2}+y^{2}\right)^{-1} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}

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