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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.18 Distributions

Terence Tao

A function is normally defined to be an object f : X → Y which assigns to each point x in a set X, known as the domain, a point f (x) in another set Y, known as the range (see THE LANGUAGE AND GRAMMAR OF MATHEMATICS [I.2 §2.2]). Thus, the definition of functions is set-theoretic and the fundamental operation that one can perform on a function is evaluation: given an element x of X, one evaluates f at x to obtain the element f(x) of Y.

However, there are some fields of mathematics where this may not be the best way of describing functions. In geometry, for instance, the fundamental property of a function is not necessarily how it acts on points, but rather how it pushes forward or pulls back objects that are ...

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