It is commonly written that the Schwarz theorem is valid for a function defined on an open set $U \subset \mathbb{R}^n$, for $n \ge 1$. But can it be valid on a closed set?
If the function is two times differentiable at the points where the definition set is closed, cannot it be correct? We would study the case "on a single side" of the definition set.
For example, if this function is two times differentiable at the point $(0,0)$:
\begin{align*} f : [0;+\infty[ \, \times \, [0;1[ ~&\longmapsto ~\mathbb{R}\\ (x,y) &\longrightarrow f(x,y) \end{align*}
can we write $\displaystyle \frac{\partial^2 f}{\partial x \partial y} (0,0) =\frac{\partial^2 f}{\partial y \partial x} (0,0) $? In fact, I would say no, writing may have helped me answer it. But what is behind this "no"?
I think we may write a sort of this equality in terms of "a single side" like "coming from the right".