linguistic:translation, linguistic:voice_change, structural:expansion
Definition

A real-valued function f defined on a domain X has a global (or absolute) maximum point at x, if f (x) ≥ f (x) for all x in X. Similarly, the function has a global (or absolute) minimum point at x, if f (x) ≤ f (x) for all x in X. The value of the function at a maximum point is called the maximum value of the function, denoted max (f (x)) {\displaystyle \max(f(x))}, and the value of the function at a minimum point is called the minimum value of the function, (denoted min (f (x)) {\displaystyle \min(f(x))} for clarity). Symbolically, this can be written as follows:

The definition of global minimum point also proceeds similarly.

If the domain X is a metric space, then f is said to have a local (or relative) maximum point at the point x, if there exists some ε > 0 such that f (x) ≥ f (x) for all x in X within distance ε of x. Similarly, the function has a local minimum point at x, if f (x) ≤ f (x) for all x in X within distance ε of x. A similar definition can be used when X is a topological space, since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows:

The definition of local minimum point can also proceed similarly.

In both the global and local cases, the concept of a strict extremum can be defined. For example, x is a strict global maximum point if for all x in X with x ≠ x, we have f (x) > f (x), and x is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x with x ≠ x, we have f (x) > f (x). Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points.

A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed and bounded interval of real numbers (see the graph above).