Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 339 / 1 / a / Solution

A vector $g$ is a subgradient of the convex function $f$ at $x$ whenfor every $y$. The set of all such vectors is the subdifferential $\partial f(x)$. The proximal operator satisfiesMore generally, $u={\mathrm{prox}}_{tf}(x)$ exactly when $(x-u)/t\in \partial f(u)$.