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(abbr. Sup). This is the least upper bound on a (real-valued) function over (a subset of) its domain. If LaTeX: f is unbounded from above, LaTeX: \textstyle \sup \{f\} = \infty, and if the domain is empty, LaTeX: \textstyle \sup \{f \} = -\infty. Otherwise, suppose U is any upper bound: LaTeX: \textstyle f(x) \le U for all LaTeX: x \in X. LaTeX: U is a least upper bound if, for any LaTeX: e > 0, there exists LaTeX: x in the domain for which LaTeX: \textstyle f(x) \ge U-e. (That is, we can get arbitrarily close to LaTeX: U in the range of LaTeX: f.) Note that the supremum is always defined, and its range is in the extended reals. The supremum is the maximum, if it is attained by some point in its domain.

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