Symmetry of one-sided partial derivatives

Question

Consider some $f: [0,1)\times [0,1)\to \mathbb{R}$. I'm interested in conditions that guarantee that the following one-sided second partial derivatives at $(x,y)=(0,0)$ are symmetric: $$ \partial_x^+ \partial_y^+ f(x,y)= \partial_y^+ \partial_x^+ f(x,y). $$ (where as usual $\partial_x^+$ and $\partial_y^+$ indicate partial derivatives defined via one-sided limits).

Everything on the topic that I've found (such as Rudin Theorem 9.41, etc.) considers $f$ defined on an open set, where regular (not one-sided) derivatives can be defined.

Is there a set of simple condition for the above equality to hold? Is there a reference for this?

(My thought right now is to "extend" $f$ to an open set $E \supset [0,1]\times[0,1]$ in a way that preserves the derivatives.)

[Cross-posted from math.stackexchange here]

Answer 1

One such condition is that $f$ be absolutely continuous in $[0,h)^2$ for some $h\in(0,1)$ -- so that $$f(x,y)+f(0,0)-f(x,0)-f(0,y)=\int_0^x du\,\int_0^y dv\,g(u,v)$$ for some function $g$ integrable on $[0,h)^2$ and for all $(x,y)\in[0,h)^2$ -- with $g$ continuous on the set $([0,h)\times\{0\})\cup(\{0\}\times[0,h))\subset[0,h)^2$. (In particular, it suffices that $\partial_x\partial_y f$ exist on $(0,h)^2$ and admit a continuous extension to $[0,h)^2$.)

Indeed, then, by dominated convergence, for all $x\in[0,h)$ $$(\partial^+_y f)(x,0)=(\partial^+_y f)(0,0)+\int_0^x du\,g(u,0)$$ and hence $$(\partial^+_x \partial^+_y f)(0,0)=g(0,0).$$ Similarly, $(\partial^+_y \partial^+_x f)(0,0)=g(0,0)$. So, $(\partial^+_x \partial^+_y f)(0,0)=(\partial^+_y \partial^+_x f)(0,0)$, as desired.

Answer 2

The following two papers by Dzagnidze appear sufficiently relevant to be worth a look:

Omar P. Dzagnidze, Unilateral in various senses: the limit, continuity, partial derivative and the differential for functions of two variables, Proceedings of A. Razmadze Mathematical Institute 129 (2002), 1−15.

Omar P. Dzagnidze, Some new results on the continuity and differentiability of functions of several real variables, Proceedings of A. Razmadze Mathematical Institute 134 (2004), 1−138.