a function with differentiable partial derivatives but unequal mixed derivatives I am looking for an example of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are both differentiable at some point, say the origin, but $\frac{\partial^2 f}{\partial x\partial y}\neq\frac{\partial^2 f}{\partial y\partial x}$ at that point.
If $f_{xy}$ and $f_{yx}$ are continuous, then they are necessarily equal.
The only examples you will find where the mixed partial derivatives are $not$ equal will be just like the standard example, $f(x,y)=x \cdot y\cdot \displaystyle \frac{x^2-y^2}{x^2+y^2}$ (and $f(0,0)=0$), and this example works precisely because $f_{xy}$ and $f_{yx}$ are not continuous at the origin. In this case $f_{xy}(0,0)=1$ and $f_{yx}(0,0)=-1$. At all other points, the mixed partials are equal.
Let $z = \arctan(y/x)$, then $\frac{d}{dy}\left(\frac{dz}{dx}\right) \not= \frac{d}{dx}\left( \frac{dz}{dy}\right)$