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

Domain, co-domain and function types

A function assigns exactly one result to each permitted input, and its type describes the sets those inputs and results belong to.

Both sets are subsets of objects in a data type. A domain might be all integers, a limited set of integers or the lower-case characters. A co-domain states the permitted results; the range is the set of results the function actually produces. The range can be smaller than the co-domain.

For example, let $A=\{-3,-2,-1,1,2,3\}$, $B=\{0,1,2,\ldots,10\}$ and $s(x)=x^2$. Then $s:A\to B$, but its range is only $\{1,4,9\}$. Every input has a result in $B$; it is unnecessary to produce 0, 2 or every other member of $B$. Several inputs may share a result: $s(-2)=s(2)=4$. That still defines a function.

The alphabet mapping $p:\{a,b,\ldots,z\}\to\{0,1,\ldots,25\}$, where $p(a)=0$, $p(b)=1$ and so on, does use every co-domain member. The rule and chosen domain decide the range.

C# represents a function value with a delegate type such as Func<char, int>. The final type parameter is the result type; earlier parameters are argument types. A programmer must still state a narrower permitted input set when the C# type alone does not express it.

using System;

class Program
{
    static void Main()
    {
        // Domain: lower-case letters a to z only.
        Func<char, int> position = letter => letter - 'a';
        Console.WriteLine(position('d')); // 3
    }
}

char permits many characters, but this mapping's stated domain does not. Calling it with 'A' violates the contract; the declared C# type is not evidence that every character is a valid alphabet input. Similarly, machine int has a finite range, so this implementation is not a function over unlimited mathematical integers.

First-class function values

A first-class function can be manipulated as a value instead of only being called by a fixed name.

Passing doubleValue passes the function; passing doubleValue(5) first applies it and passes the result 10. The distinction matters because a receiving function can choose when and to which value it applies the supplied operation.