Proper extended-real function

An extended-real function is proper if it never takes $-\infty$ and is finite at at least one point. It can take $+\infty$ to impose a constraint. A convex function with this property is a proper convex function.

The effective domain of an extended-real function is $\mathrm{dom}E=\{u:E(u)<+\infty \}$. An infinite value can impose a hard constraint through an indicator functional of a constraint set.