Upper and lower bounds
Bounds of functions

The definitions can be generalized to functions and even to sets of functions.

Given a function f with domain D and a preordered set (K, ≤) as codomain, an element y of K is an upper bound of f if y ≥ f (x) for each x in D. The upper bound is called sharp if equality holds for at least one value of x. It indicates that the constraint is optimal, and thus cannot be further reduced without invalidating the inequality.

Similarly, a function g defined on domain D and having the same codomain (K, ≤) is an upper bound of f, if g (x) ≥ f (x) for each x in D. The function 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 ≤.
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