section_path: Notation/Arrow notation
section_name: Arrow notation
paragraph_text: 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, x ↦ x + 1 {\displaystyle x\mapsto x+1} is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of R {\displaystyle \mathbb {R} } is implied.
  
  The domain and codomain can also be explicitly stated, for example:
  
  This defines a function sqr from the integers to the integers that returns the square of its input.
  
  As a common application of the arrow notation, suppose f: X × X → Y; (x, t) ↦ f (x, t) {\displaystyle f:X\times X\to Y;\;(x,t)\mapsto f(x,t)} is a function in two variables, and we want to refer to a partially applied function X → Y {\displaystyle X\to Y} produced by fixing the second argument to the value t without introducing a new function name. The map in question could be denoted x ↦ f (x, t 0) {\displaystyle x\mapsto f(x,t_{0})} using the arrow notation. The expression x ↦ f (x, t 0) {\displaystyle x\mapsto f(x,t_{0})} (read: "the map taking x to f of x comma t nought") represents this new function with just one argument, whereas the expression f (x, t) refers to the value of the function f at the point (x, t).