The Proximal Map of a Convex Function on $\mathbb{R}^n$

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers and the Euclidean norm $\lVert\,\cdot\,\rVert$ of $\mathbb{R}^{n}$ are as fixed there.

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be convex on $\mathbb{R}^{n}$, which is a convex subset of itself. For $x\in\mathbb{R}^{n}$ let $\phi_{x}:\mathbb{R}^{n}\to\mathbb{R}$ be given by $\phi_{x}(z)=f(z)+\tfrac{1}{2}\lVert x-z\rVert^{2}$; by Existence and Uniqueness of the Proximal Minimiser of a Convex Function (i) there is exactly one point of $\mathbb{R}^{n}$ at which $\phi_{x}$ attains its least value on $\mathbb{R}^{n}$, so the following assigns to each $x$ a well-determined point.

Definition. (i) The proximal map of $f$ is the map $J_{f}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ sending each $x\in\mathbb{R}^{n}$ to the unique $y\in\mathbb{R}^{n}$ such that $\phi_{x}(y)\le\phi_{x}(z)$ for every $z\in\mathbb{R}^{n}$. When $f$ is clear from the context we write $J$ for $J_{f}$.