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Does anyone know of a counterexample (or proof) for the following?

Suppose $f: D \to \mathbb{R}, D \subset \mathbb{R}^2$ is continuous at $(a,b)$ and its directional derivatives are linear in the components of the unit-length direction vector $u = (u_1,u_2)$ (i.e., there are finite constants $c_1$ and $c_2$ such that for any $u$, $D_u f(a,b) = c_1 u_1 + c_2 u_2)$. Then $f$ is differentiable at $(a,b)$.

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1 Answer 1

Counterexample: $$f(x,y)=\begin{cases} \frac{x^2-|y-2x^2|}{|x|} \quad & \text{if }x^2<y<3x^2 \\ 0 \quad &\text{otherwise} \end{cases} $$ Since $0\le f(x,y)\le |x|$ for all $(x,y)$, we have continuity.

Restricting $f$ to any line through the origin, we find that the restriction is identically zero in some neighborhood of the origin. Hence all directional derivatives at $(0,0)$ are $0$.

Yet, $f(x,2x^2)=|x|$, which prevents $f$ from being differentiable at $(0,0)$.

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