I've seen proof that a signed distance function $f$ satisfies the eikonal equation
$$\|\nabla_{\mathbf{x}} f(\mathbf{x})\| = 1, \mathbf{x} \in \Omega$$
where $\Omega$ is an open set with boundary $\partial \Omega$ such that $f|_{\partial\Omega} = 0$, and $\|\cdot\|$ is the Euclidean norm.
However, the main Wikipedia article for the eikonal equation makes a stronger claim than this: it states that the solution to this equation is the signed distance function.
I've seen similar claims in some papers; e.g., that the signed distance function is the viscosity solution to this eikonal equation. However, I haven't seen any lucid proofs of this fact.
So, is unit norm gradient equivalent to being a signed distance function?
Any help is appreciated.