Analysis II | Part IB, 2001

Prove that if all the partial derivatives of f:RpRf: \mathbb{R}^{p} \rightarrow \mathbb{R} (with p2p \geqslant 2 ) exist in an open set containing (0,0,,0)(0,0, \ldots, 0) and are continuous at this point, then ff is differentiable at (0,0,,0)(0,0, \ldots, 0).


g(x)={x2sin(1/x),x00,x=0g(x)= \begin{cases}x^{2} \sin (1 / x), & x \neq 0 \\ 0, & x=0\end{cases}


f(x,y)=g(x)+g(y).f(x, y)=g(x)+g(y) .

At which points of the plane is the partial derivative fxf_{x} continuous?

At which points is the function f(x,y)f(x, y) differentiable? Justify your answers.

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