Non-continuous higher differentiability, II

Question

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.

1. Does strong twice differentiability at $x$ imply that $f$ is differentiable at $x$?
2. 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 $U$?
3. Can you give any reference for a definition such as this?

Answer 1

1. No. Think of a $\mathbb Q$-linear map $f:\mathbb R\to \mathbb Q$.

2. Yes. For small nonzero $v$ and $w$ write $$ E(v,w)=\frac{f(a+v+w)-f(a+v)-f(a+w)+f(a)-b(v,w)}{|v||w|}. $$

By assumption you have bilinear $b$ such that the limit of $E(v,w)$ as $(v,w)\to (0,0)$ is zero. If you also assume that $f$ is differentiable at $a+v$ for all small enough $v$ then for small nonzero $v$ you can take the limit of $E(v,w)$ as $w\to 0$ and get $$ \frac{df_{a+v}(\frac{w}{|w|})-df_{a}(\frac{w}{|w|})-b(v,\frac{w}{|w|})}{|v|}. $$ The limit of this as $v\to 0$ is zero. Therefore the derivative of $x\mapsto df_x$ at $x=a$ exists and corresponds to $b$.

Note that this implies that you get continuity of $df$ at a point if you have existence of $df$ in a neighborhood of the point and existence of $d^2f$ (in the sense we are exploring) at the point.