In a comment on this question, Tom Goodwillie proposed a notion of higher differentiability that I elaborate to something like the following:

Let $f:\mathbb{R}^n \to \mathbb{R}$. Let's say that $f$ is *strongly twice differentiable* at $x$ if there is a bilinear map $d^2 f_x:\mathbb{R}^n\times \mathbb{R}^n \to \mathbb{R}$ such that

$$ f(x+v+w) - f(x+v) - f(x+w) + f(x) = d^2f_x(v,w) + E(v,w){|v|}{|w|} $$

where $\lim_{v,w\to 0} E(v,w) = 0$. He pointed out that such a $d^2 f_x$, if it exists, is symmetric, since the LHS above is symmetric in $v$ and $w$. On the other hand, the usual proof of equality of mixed partials essentially shows that if $f$ is $C^2$ in a neighborhood of $x$, then it is strongly twice differentiable in this sense.

- Does strong twice differentiability at $x$ imply that $f$ is differentiable at $x$?
- If $f$ is differentiable in a neighborhood $U$ of $x$, does strong twice differentiability at $x$ imply that $f$ is twice differentiable in the usual sense, i.e. that $d f:U \to L(\mathbb{R}^n,\mathbb{R})$ is differentiable at $x$? Do we need $f$ to be strongly twice differentiable on a whole neighborhood $U$?
- Can you give any reference for a definition such as this?