Editing Upper and lower bounds (section)

==Bounds of functions==

The definitions can be generalized to [[Function (mathematics)|functions]] and even to sets of functions.

Given a function {{italics correction|{{mvar|f}}}} with [[Domain of a function|domain]] {{mvar|D}} and a preordered set {{math|(''K'', ≤)}} as [[codomain]], an element {{mvar|''y''}} of {{mvar|K}} is an upper bound of {{italics correction|{{mvar|f}}}} if {{math|''y'' ≥ {{italics correction|''f''}}(''x'')}} for each {{mvar|x}} in {{mvar|D}}. The upper bound is called ''[[Mathematical jargon#sharp|sharp]]'' if equality holds for at least one value of {{mvar|x}}. It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality.

Similarly, a function {{mvar|g}} defined on domain {{mvar|D}} and having the same codomain {{math|(''K'', ≤)}} is an upper bound of {{italics correction|{{mvar|f}}}}, if {{math|''g''(''x'') ≥ {{italics correction|''f''}}(''x'')}} for each {{mvar|x}} in {{mvar|D}}. The function {{mvar|g}} is further said to be an upper bound of a set of functions, if it is an upper bound of ''each'' function in that set.

The notion of lower bound for (sets of) functions is defined analogously, by replacing ≥ with ≤.