Suppose, in a 2D plane, there is an electric field: $$\vec{E}(x,y)=E_0\begin{pmatrix}-y \\ +x\end{pmatrix}=E_0(-y\vec{e}_x+x\vec{e}_y)$$ $$\vec{E}(r,\theta)=E_0 r \vec{e}_\theta$$ where $r,\theta$ are polar coordinates. This electric field goes around in a circle and closes in itself. What would the potential, defined through $\vec{E}=-\nabla V$, be as a function of $\theta$ at a given radius from the origin?
If we define the $-\infty<\theta<+\infty$, it would be a simple linear function in $\theta$: $V(r,\theta)=-E_0 r \theta$. But if we restrict $0\leq\theta<2\pi$, surely it would become a matter of convention, whether it is a single-valued function (therefore leading to a discontinuity in $\vec{E}$) or a multi-valued function?