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Use the extended-real characteristic functionalFor $\eta <0$ its sublevel set is empty, while for every finite $\eta \ge 0$ its sublevel set is $X^{\prime }$. Both are closed when $X^{\prime }$ is closed, so part ii proves that this indicator functional of a constraint set is lower semicontinuous.

For a nonempty closed convex set $C$, the squared-distance function is the Moreau envelope of its indicator functional of a constraint set. It is convex and differentiable withand this gradient is one-Lipschitz.