While working on some properties of partial derivatives and multiply differentiable functions of several variables, I came across the following Hypothesis 1:
Let $f: \mathbb{R}^n\to\mathbb{R}$, $\delta>0$, $k\in \mathbb{N}$ and for each partial derivative $D^{k-1}f:=\partial_{i_1}\ldots\partial_{i_{k-1}}f$ of $f$ of order $k-1$ one of its partials $\partial_1D^{k-1}f,\ldots,\partial_nD^{k-1}f$ exists at $\mathbb{a}\in \mathbb{R}^n$ and that the remaining $n-1$ of them exist in some neighborhood $O_\delta(\mathbb{a})$ and are continuous at $\mathbb{a}$. Then $f$ is $k$ times differentiable at $\mathbb{a}$ (i.e. $f$ is $k-1$ times differentiable in some neighborhood of $\mathbb{a}$ and all of its partials of order $k-1$ are differentiable at $\mathbb{a}$).
For $k=1$ the statement is true and can be found, for example, in T.Apostol's "Mathematical Analysis" (Th. 12.11). I wonder if anyone saw this statement somewhere for $k\geqslant 2$ or has any suggestions on its proof (or maybe a counterexample for disproof)?
First I started with the case $n=k=2$ and tried to find a counterexample (similar to one constructed in https://www.mdpi.com/2227-7390/8/11/1946/htm), namely I tried to construct a function $f(x,y)$ such that $f_{xy}$ , $f_{yx}$ exist at $(0,0)$ and $f_{xx}$, $f_{yy}$ exist in some neighborhood and are continuous at $(0,0)$, but $f$ is $\textit{not}$ twice differentiable at $(0,0)$ (i.e. there is no neighborhood of $(0,0)$ where $f$ is differentiable, because the defferentiability of $f_x$ and $f_y$ at $(0,0)$ follows from the Apostol's case $k=1$). But T.Tao pointed out that in this setting $f$ belongs to $C^1$ in some neighborhood of $(0,0)$ by Shauder estimates (see A non-differentiable function $f(x,y)$ with bounded $f_x$, $f_y$, $f_{xx}$ and $f_{yy}$), so $f$ is in fact $\textit{twice differentiable}$ at $(0,0)$! Now it seems to me that Hypothesis 1 is in fact true! In order to prove it in case $n=k=2$ there are two cases remaining: assume that $f_x$ and $f_y$ are differentiable at $\mathbb{a}$ and either
- $f_{xx}$ and $f_{yx}$ exist in some neighborhood and are continuous at $(0,0)$, or
- $f_{xy}$ and $f_{yx}$ exist in $O_\delta(\mathbb{a})$ exist in some neighborhood and are continuous at $(0,0)$.
Prove that $f$ is differentiable in some neighborhood of $(0,0)$.
Shauder estimates don't work in these cases. Any suggestions?
