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Do there exist two functions $u,v$ defined on an open set containing $(0,0)$ with values in $\mathbb{R}$ such that

(1) $u,v$ are differentiable at $(0,0)$;

(2) $u_x=v_y, u_y=-v_x$ at $(0,0)$;

(3) at least one of $u_x,u_y,v_x,v_y$ is not continuous at $(0,0)$ ?

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up vote 1 down vote accepted

If you take $u(x, y) = v(x, y) = (x^2 + y^2)\, f(x, y)$, where $f$ is bounded, the first two conditions are already satisfied, since the derivatives are $0$.

To get discontinuous derivatives, you only have to make $f$ oscillate enough around $(0, 0)$, for example $f(x, y) = \sin \frac{1}{x^2 + y^2}$.

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