Editing Function (mathematics) (section)

=== Arrow notation ===
Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. It uses the ↦ arrow symbol, pronounced "[[maps to]]". For example, <math>x\mapsto x+1</math> is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of <math>\R</math> is implied.

The domain and codomain can also be explicitly stated, for example:

<math display="block">\begin{align}
\operatorname{sqr}\colon \Z &\to \Z\\
x &\mapsto x^2.\end{align}</math>

This defines a function {{math|sqr}} from the integers to the integers that returns the square of its input.

As a common application of the arrow notation, suppose <math>f: X\times X\to Y;\;(x,t) \mapsto f(x,t)</math> is a function in two variables, and we want to refer to a [[Partial application|partially applied function]] <math>X\to Y</math> produced by fixing the second argument to the value {{math|''t''<sub>0</sub>}} without introducing a new function name.  The map in question could be denoted <math>x\mapsto f(x,t_0)</math> using the arrow notation. The expression <math>x\mapsto f(x,t_0)</math> (read: "the map taking {{mvar|x}} to {{mvar|f}} of {{mvar|x}} comma {{mvar|t}} nought") represents this new function with just one argument, whereas the expression {{math|''f''(''x''<sub>0</sub>, ''t''<sub>0</sub>)}} refers to the value of the function {{mvar|f}} at the {{nowrap|point {{math|(''x''<sub>0</sub>, ''t''<sub>0</sub>)}}.}}