# Infimum

### From Glossary

(abbr. *Inf*). The greatest lower bound on a (real-valued)
function over (a subset of) its domain. If f is unbounded from
below, Inf{f} = -infinity, and if the
domain is empty, Inf{f} = infinity. Otherwise, suppose L is any
lower bound: f(x) >= L for all x in X. L is a *greatest*
lower bound if, for any e > 0, there exists x in the domain for
which f(x) <= L+e. (That is, we can get arbitrarily close to L
in the range of f.) Note that the infimum is always defined, and
its range is in the extended reals. The infimum is the minimum, if it is attained by
some point in its domain.