Differentiable with discontinuous partial derivatives I am looking for examples of functions defined on $\mathbb R^n$ which are differentiable but whose partial derivatives are not continuous. In the reals I guess the simplest example is $x^2 \sin{1/x}$ but the generalisation of this to higher dimensions is too messy for me (e.g. $(x^2+y^2)\sin{\frac{1}{\sqrt{x^2+y^2}}}$) and I was wondering if there are nicer examples (particularly in $\mathbb R^2$)

If you have a one-dimensional example, it automatically produces higher dimensional ones. Let 
$$
f(x)=\begin{cases} x^2\sin\frac1x, & x\ne 0, \\ 0, & x=0,\end{cases}$$
and define 
$$
F(x_1, x_2, \ldots, x_n)=f(x_1).$$
If you want all partial derivatives to be discontinuous, then define
$$
F(x_1, x_2, \ldots, x_n)=f(x_1)+ f(x_2)+ \ldots+f(x_n).$$
