What is the mathematical definition of a function?

Question

What is the mathematical definition of a function?

Answer 1

The definition is:

A function is a relation between two sets A and B so that
each element of A has a correspondent, and only one, in B

Comments:
1) as each number of A has a correspondent... then all elements of A must be used... a element of A cannot rest unused.
2) the correspondent in B must be unique for each element in B, but the elements of B can be used more than one time... and someone may rest unused.

Answer 2

Informally, a function is a mapping from one set to another such that every element in the first set maps to only one element in the second set. It may be that multiple elements in the first set map to the same element in the second set, but it cannot be that multiple elements in the second set are mapped to by the same element in the first set. The set of elements from the first set in these mappings is known as the domain. The set of elements mapped to from the domain is known as the range.

A little more formally, a function is a set of ordered pairs (a,b) such that if (a,b) and (a,c) are both in the set, then b=c. The domain is then the set of all such a and the range is the set of all such b.

Sometimes there will be a notion of "codomain" which is often thought of as "range." A function that maps to a "codomain" may not have every element of the codomain in its map. In other words, the range is a subset of the codomain.

Answer 3

For one value of x there is only one value of y then it is said to function.

Answer 4

informal: it has to be continuous, differentiable at all points and have only one y value for every x value

example: The letter "V" (at the point) has one y for that x value, but the letter "V" is NOT a function!

Answer 5

a relationship or expression involving one or more variables