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I am trying to come up with a function $g:\mathbb{R}^{2} \to\mathbb{R}$ which is differentiable at each point $(x,y)$ in $\mathbb{R}^{2}$ but whose partial derivatives are not continuous at $(0,0)$. Can anyone give me examples of such functions?

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up vote 5 down vote accepted

From Counterexamples in Analysis, Gelbaum and Olmsted , page 119 $$f(x,y)=\cases{ x^2\sin(1/x)+y^2\sin(1/y),&$xy \ne 0$\cr x^2\sin(1/x), &$x \ne 0, y=0$ \cr y^2\sin(1/y), &$x=0, y\ne0$ \cr 0,&$x=y=0$ }$$

The following may be helpful:

I believe, but haven't proved, that if you take the graph of $$g(x)=\cases{x^2\sin(1/x), &$x\ne0$ \cr 0,&$x=0$ }$$ in the $x$-$z$-plane and "spin the right half of it about" the $z$-axis, you'll obtain an example of the function you want. At any rate, this captures the flavor of the Gelbaum and Olmsted example (but would be harder to work with).

Note that $$g'(x)=\cases{2x\sin(1/x)-\cos(1/x),&$x\ne0$\cr0,&$x=0$ };$$ so, $g'$ is discontinuous at $x=0$.

enter image description here

$\ \ \color{darkgreen}{z= g(x)},\quad \color{maroon}{z=x^2},\quad\color{darkblue}{z=-x^2}$

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