I was thinking about Young's and Schwarz's theorem on when do partial derivatives be equal and I was wondering about how smooth can a function whose mixed partial are not equal be. I was wondering there is a function $f:\mathbb{R}^2 \to \mathbb{R}$, such that the following hold.
a) $f$ is continuous in $\mathbb{R}^2 $ (We can assume $f(0,0) = 0$)
b) $f_x$ and $f_y$ and their partial derivativess exist everywhere (We can assume $f_x(0,0) = f_y(0,0) = 0$)
c) $f_x$ and $f_y$ are continuous in $\mathbb{R}^2$ (so $f$ is differentiable at $(0,0)$) and the directional derivatives at $(0, 0)$ can be expressed as $L(v)$, where $L$ is linear.
d) $f_{xx}$, $f_{yx}$, $f_{xy}$, $f_{yy}$ are continuous in $\mathbb{R}^2\smallsetminus\{(0, 0)\}$.
e) $f_{xx}$ is continuous at $(0,0)$ (so $f_x$ is differentiable at $(0, 0)$). (We can assume $f_{xx}(0,0) = f_{yy}(0,0) = 0$.
If $f_y$ was also differentiable at $(0, 0)$, by Young's theorem mixed partial would be equal. If $f_{yy}$ or $f_{xy}$ we continuous at $(0,0)$, $f_y$ would be differentiable at ($0, 0$). Assumptions can be made because if such and $f$ exist, then $f - f(0,0)- xf_x(0,0) - yf_y(0,0) - \frac{x^2}{2}f_{xx}(0,0) - \frac{y^2}{2}f_{yy}(0,0)$ will also satisfy the desired property.
