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Partial functions are defined similarly to ordinary functions, with the total condition removed Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above. That is, a partial function from x to y is a binary relation r between x and y such that, for every there is at most one y in y such that using functional notation, this means that, given either is in y, or it is undefined

The set of the elements of x such that is defined and belongs to y is. A point where a function is discontinuous is called a discontinuity The latter notation is especially useful in combination with the currying isomorphism

Given a function , its application is represented as by the former notation and (or with the argument written with the less common angle brackets) by the latter

However, functions in curried form can be represented by juxtaposing their arguments In mathematics, a function space is a set of functions between two fixed sets Often, the domain and/or codomain will have additional structure which is inherited by the function space For example, the set of functions from any set x into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication

In other scenarios, the function space might. A function f from x to y The set of points in the red oval x is the domain of f It is sometimes denoted by or , where f is the function

Notation and terminology the notation is also used to denote the characteristic function in convex analysis, which is defined as if using the reciprocal of the standard definition of the indicator function

A related concept in statistics is that of a dummy variable. The formal definition of an iterated function on a set x follows Let x be a set and f X → x be a function

And where idx is the identity function on x and (f g) (x) = f (g(x)) denotes function composition This notation has been traced to and john frederick william herschel in 1813 If the first is the name (symbol) of a function, denotes the value of the function applied to the expression between the parentheses For example, the functions and are discontinuous at 0, and remain discontinuous whichever value is chosen for defining them at 0

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