A _map_, _mapping_, or a _function_, $f$ is a [mathematical object](Mathematical%20object.md) that associates members of two [sets](Sets.md) or [subsets](Sets.md#subsets) with each other^[The terms _map_, _mapping,_ and _function_ always refer to the same thing, however, in a lot of literature, the term _function_ is preferred when describing maps between sets of [numbers,](Numbers%20(index).md) while the term, map, may be considered more general or only used in contexts with other [mathematical objects.](Mathematical%20object.md) Additionally, when conceptualizing functions as abstract machines that take one input and give one output rather than as a relation between [sets,](Sets.md) the term, function, is more common. This is why in [analysis](Analysis%20(index).md) we tend to use the word 'function' while [algebra](Algebra%20(index).md) and [geometry](Geometry%20(index).md) tend to use the words 'map' and 'function' in different contexts] It is a rule, $r,$ that assigns one member of one set, $A,$ to a member of another set, $B$ (i.e. a [rule of assignment](Rule%20of%20assignment.md)) along with the set $B.$^68c6a1 A [map](Maps.md), $f,$ between [sets](Sets.md) $A$ and $B$ it is written as $f:A \rightarrow B$ where $a \in A$ and $b \in B.$ The [set](Sets.md) $A,$ in [the definition of a map](Maps.md#^68c6a1) is the _domain of a map_ or _domain of a function_ as well as the [domain its rule of assignment,](Rule%20of%20assignment.md#Domain%20of%20a%20rule%20of%20assignment) $r.$ We refer to [set,](Sets.md) $B$ in [the definition of a map](Maps.md#^68c6a1) as the _range of a map,_ or _range of a function,_ where the range itself is denoted as $f(a)$ for all $a\in A.$ The range of a function is also the [image of its rule of assignment,](Rule%20of%20assignment.md#Image%20of%20a%20rule%20of%20assignment) $r.$ Thus, $f(a),$ is also referred to as the _image set_ of $f.$ # Continuous maps [Continuity](Continuity.md) ## Smooth maps [[smoothness]] # Generalization %%It is equivalent to a _function_ in many contexts, however, this is not always the case in every area of study. More generally it is equivalent to a _morphism_, where a _morphism_ is a map between two objects in an abstract _category_.%% # partial map #MathematicalFoundations/SetTheory #MathematicalFoundations/Analysis #MathematicalFoundations/Algebra #MathematicalFoundations/Geometry