Proper convex function

A convex function f taking on values on the extended real number line is said to be proper if f(x) < ∞ for at least one x and f(x) > − ∞ for every x. This definition is motivated by the fact that the extended real number line does not constitute a field since e.g. the value of the expression ∞ − ∞ is left undefined.

It is always possible to consider the restriction of a proper convex function f to its effective domain

${\mbox{dom}}f=\left\{x:f(x)<\infty \right\}$

instead of f itself, thereby avoiding some minor technicalities that may otherwise arise. The effective domain of a convex function is always a convex set.

References

• Rockafellar, Ralph Tyrell, Convex Analysis, Princeton University Press (1996). ISBN 0691015864