= Solution
The <forward subgradient step> and <backward subgradient step> are the set-valued maps
$$
\boxed{
F_{\tau f}(x)=x-\tau\partial f(x)
=\{x-\tau p:p\in\partial f(x)\},\qquad
B_{\tau f}=(I+\tau\partial f)^{-1}.}
$$
Thus $u\in B_{\tau f}(z)$ means $z-u\in\tau\partial f(u)$. They are respectively the explicit and implicit time discretizations of <subgradient flow> $\dot x\in-\partial f(x)$. At a differentiable point the forward update is $x-\tau\nabla f(x)$, while the backward update evaluates the gradient at the new point. For proper closed convex data the latter is the <proximal operator>
$$
B_{\tau f}(z)=\arg\min_u\left\{f(u)+\frac1{2\tau}\|u-z\|^2\right\}.
$$
To prove at most one output, suppose $u,v\in B_{\tau f}(z)$. Then $(z-u)/\tau\in\partial f(u)$ and $(z-v)/\tau\in\partial f(v)$. The two <subgradient inequalities> imply <monotonicity of a convex subdifferential>, giving
$$
0\leq\left\langle\frac{z-u}{\tau}-\frac{z-v}{\tau},u-v\right\rangle
=-\frac{\|u-v\|^2}{\tau}.
$$
Since $\tau>0$, $u=v$. Convexity supplies this monotonicity; lower semicontinuity is not needed for this at-most-one argument.
Under the full printed assumptions the output actually exists at every $z$. Properness ensures a finite point and excludes $-\infty$. Proper lower-semicontinuous convex $f$ has an <affine minorant>, so the quadratic proximal objective is coercive. Lower semicontinuity makes its minimum attained on a compact sublevel set. Convexity and the positive quadratic curvature make it strongly convex, hence its minimizer is unique. The <subgradient optimality condition>, with the differentiable quadratic term, is exactly $z-u\in\tau\partial f(u)$. Thus
$$
\boxed{B_{\tau f}\text{ is everywhere defined and single-valued}.}
$$
This also identifies precisely which assumptions provide existence, uniqueness and the update's optimality interpretation.
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