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The standard absolute value function on the real numbers.
An absolute value is the function defined by
It is defined using the order axioms on and measures distance via , a basic example of a metric. Absolute value is built into – definitions such as the limit at a point.
Order axioms: Let be a field (satisfying the field axioms) and let be a relation on . The following are required for all :
A metric on a set is a function such that for all :
A metric is the basic structure underlying a metric space; it determines open balls and hence the metric-induced topology.
A limit at a point for a function (with ) at a point is a number such that: for every there exists with the property that whenever and , one has . Typically one assumes that is a limit point of .
The inequalities use the absolute value to measure distance on the real line. One-sided variants are captured by the one-sided limit.