Exhibit a function $f(x,y): \mathbb R^2 \to \mathbb R$ such that all directional derivatives equal $0$ at the origin, yet the function is unbounded in any neighborhood of the origin.
My work is below, to which I request verification, suggestions, and feedback, both on the proof and exposition. In particular: I tried to write the proof so it would be succinct and intuitive, and yet still rigorous. Was this accomplished? Could the exposition be improved? What about the geometrical remark?
Note: Examples and proofs exist; this question is to verify this example and proof.
The function $$f(x,y) = \begin{cases} 0 \text{ if } y \leq 0 \text{ or } y \geq x^2 \\ \frac 1 x \text{ otherwise.} \end{cases}$$ meets the criteria.
Proof: By definition, $$D_vf(0,0) = \lim_{h \to 0} \frac {f(hv_x, h v_y) }{h}.$$ The function $\frac {f(hv_x, hv_y)}h$ equals $0$ unless$$0 < h v_y < h^2v_x^2 \tag{1}$$ but as $h \to 0$, $$\frac {h^2v_x^2}{hv_y} = hk \to 0$$ for all $k$ and $(1)$ is not met. Thus $D_vf(0,0) = 0$.
We now show the function is unbounded in any ball around the origin. For all $\delta > 0, L \in \mathbb R$, there exists $(x,y) \in B_\delta(0,0)$ such that $f(x,y) > L$. If $L \leq 0$, this is trivial; otherwise, let $x = \frac 1 {2L}$ and $y = x^2/2$.
Remark: This result is essentially a corollary from Euclid's Elements III.16 and III.2, except that Euclid uses a circle instead of a parabola. Euclid shows if $\ell$ is a line touching point $P$ of circle $O$, and $\ell$ is not $\perp OP$, then $\ell$ also touches circle $O$ at another point $Q, Q \neq P$ [III.16] and that the segment $PQ$ is entirely interior to the circle [III.2]. Thus (if we assume these results hold for parabolas as well, which they do), for any vector $v$, there exists $h \neq 0$ such that $hv$ is entirely above the parabola $y = x^2$ and therefore uniformly $0$.
Euclid also mentions, but does not explore, horn angles, which are curves that a. intersect $P$ b. do not intersect the circle anywhere besides $P$ and c. do not intersect the perpendicular to $OP$ anywhere but $P$. It is such a horn angle which we find and follow to exhibit an unbounded $(x,y)$ in any neighborhood around the origin.