4.12.1.1–4.12.1.3 - Function types, first-class values and application (Java)

Domain, co-domain and function types

A function gives exactly one result for each input in its stated domain.

For example, let $A=\{0,1,2,3\}$, $B=\{0,1,2,3,4,5,6\}$, and $f(x)=2x$. The function type is $f:A\rightarrow B$. Applying it to every member of $A$ produces $0,2,4,6$. Those actual outputs form the range, a subset of the co-domain. The unused values $1,3,5$ are still members of $B$; the function need not produce every member of its co-domain. A value such as $4$ is outside this particular domain, even though it is an integer.

Two different inputs may give the same result. For $h:\{-2,-1,0,1,2\}\rightarrow\{0,1,2,3,4\}$ with $h(x)=x^2$, both $-2$ and $2$ give $4$. That is still a function. An input being assigned two different results would break the function rule.

In Java, Function<Integer, Integer> describes a function value with Integer input and result types. The first generic type is the argument type; the second is the result type. Generic types use the object type Integer, rather than primitive int. Java's type alone does not express a finite domain such as $A$: the example's contract must also state its allowed values. Mathematical integers are unbounded; Java integer arithmetic has finite limits. All numerical examples here use small values within those limits.

First-class function values

A first-class function can be handled as a value, rather than only called by a fixed name.

Integers and strings are familiar first-class values. A function value can be selected and passed around in similar ways. A Java lambda, such as n -> n + 1, supplies the behaviour of a functional interface, an interface with one abstract method. Here Function supplies apply. The lambda expression creates a value implementing that interface; Java does not turn every ordinary method into a standalone function object automatically.

Read this complete Java 8 program. choose returns a function value. Assigning it to rule does not apply that function to an integer. applyTwice receives that value as an argument, and applies it to a value and then to the first result.