Infimum

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(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.


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