Are the following conditions necessary and sufficient for $f$ being differentiable at $x$?

Question

Let $f : U \rightarrow \mathbb{R},$ where $U \subseteq \mathbb{R}^n$ is open, and let $x \in U.$ Then is the following true or false?

$f$ is differentiable at $x$ if and only if...

1) all directional derivatives of $f$ at $x$ exist and

2) $D_{u+v}f(x) = D_{u}f(x) + D_{v}f(x)$ and $D_{cv}f(x) = cD_{v}f(x)$ for all $u, v \in \mathbb{R}^n$ and $c \in \mathbb{R}.$

I know that just the directional derivatives existing is not enough to imply differentiability. I found many answers on this site about that already, but I couldn't find any questions that included the other conditions as well. Do those extra conditions make it true?

Answer 1

No. you need uniformity of some sorts. Here is a counterexample.

Let $n=2$, $A = \{ (r \cos \theta, r \sin \theta ) | \theta \in (0, 2 \pi], 0 \le r \le \theta\}$, and define $f= 1_A$.

Then $df((0,0), h) = 0$ for all $h$, but $f$ is not differentiable at $(0,0)$.

Answer 2

These conditions are not sufficient.

For example, take any function $f: \mathbb{R}^2 \to \mathbb{R}$ which is $\mathbb{Q}$-linear but not $\mathbb{R}$-linear.