= Solution
Let $r(u)=\|u-g\|_2^2$ and fix the reference noise budget $\sigma$. Define the <convex perturbation function>
$$
f(u,z)=\operatorname{TV}(u)+\delta_{(-\infty,0]}(r(u)-\sigma-z).
$$
Positive $z$ increases the allowed squared noise level. The feasible epigraph set $r(u)\leq\sigma+z$ is convex and closed, so this is a proper <lower semicontinuous> jointly convex perturbation. Writing the inequality indicator as a supremum over its multiplier gives
$$
\boxed{\inf_u\sup_{\lambda\geq0}L(u,\lambda),\qquad
L(u,\lambda)=\operatorname{TV}(u)+\lambda(r(u)-\sigma).}
$$
The <Lagrange dual function> is $d(\lambda)=\inf_uL(u,\lambda)$ for $\lambda\geq0$. In the signed <convex conjugate> convention of part (a), $y=-\lambda$; positive $y$ gives dual objective $-\infty$ because the perturbation can be made arbitrarily large.
The feasible ball is nonempty and compact, so <lower semicontinuity> of <total variation> gives a primal minimizer. For $\sigma>0$, $u=g$ satisfies $r(g)=0<\sigma$, providing the <Slater condition>. Total variation is finite everywhere in the given discretization and hence continuous. <Strong duality> and dual attainment give a finite optimal $\lambda^*\geq0$. The pair satisfies the <Karush-Kuhn-Tucker conditions>
$$
r(u^*)\leq\sigma,\quad\lambda^*\geq0,\quad
\lambda^*(r(u^*)-\sigma)=0,\quad
0\in\partial\operatorname{TV}(u^*)+2\lambda^*(u^*-g).
$$
Equivalently,
$$
\boxed{L(u^*,\lambda)\leq L(u^*,\lambda^*)\leq L(u,\lambda^*)\quad
\text{for all }u\text{ and }\lambda\geq0.}
$$
Thus the set of saddle points is nonempty. Compactness handles primal attainment, while strict feasibility supplies a finite multiplier; these are distinct steps. An inactive constraint can yield $\lambda^*=0$. At $\sigma=0$, feasibility still forces $u=g$, but this strict-feasibility argument does not apply.
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