Yes.

There are different kinds of things that variously get called
functions:

 total functions, partial functions, multifunctions, aggregate
functions

Often "function" by itself means "total function" but sometimes it
doesn't.

It seems the difference between a total function and a partial
function is
just in what we want to call the domain. Division over the domain
(integer, integer) is partial; division over the domain (integer,
nonzero integer)
is total. What's up with that?

Some things out there produce more than one result, or a stream of
results. Sometimes these are called multifunctions and sometimes
generators.

Then we have things like sum() avg() etc. Aggregate functions.

Marshall

It does. It does. :-)

-- Jan Hidders

This is not good enough.  It is possible that a value exists in the
domain Sn yet the relation has no corresponding tuple which holds that
value for An.

This seems right.  A domain is just a set of values.  In relational
algebra, this set is required to be non-empty since attributes are
non-null.

On the other hand, mathematics does not require a domain to be non-empty.

No, there is always just one meaning of function in database.

Essentially correct, although to be rigorous you need to define how
such binary relation can define the meaning of "associating each
element in D exactly one element in C."

Not all binary relation has this property.

Yea.  It's really an abused use of the term in software design.  
Subroutines in software has no clear domain since same input arguments
can product different outputs.

Fair.

Yes.

It will be good to say something about some specifics that hold true
for a function but doesn't hold for the relations in general.

1)   The rule "f pairs each x with just one y"
For example if one step on a scale then the scale will show just one
number. The scale will not show two or three numbers in same time. We
have here the law:  weight = gravity x mass or y = gx. Because "f-
machine" compute y, we can say that y is an extrinsic property of the
person who step on the scale.
Another example for above rule is a sequence of procedures:

----- > f1----- > f2 ------ >  ...

The sequence is very important for the structural programming and for
some other sciences.

For presenting function as "f-machine" we can use:
Definition1    A function from A to B is a rule that assigns, to each
member of set A, exactly one member of set B.
Counter - example: In a theatre we can imagine a function from the set
of visitors to the set of seats in the theatre. However we can't say
what is the rule here. This is weak side of Definition1. We can use
set-theoretic Definition2, formally it is OK for this example, but
still we don't know what the rule here is. As far as I know there is
no good definition for the algorithm; usually the algorithm is defined
as the fixed set of rules.

2) Construction of a relation
(i) For example the following relation R ( identifier1, A1, A2)  -
where identifier1 get values from a procedure P, and D1, D2 are
domains for A1, A2 - can't be constructed as the product of P x D1 x
D2
because P is not set, it is the procedure. So to construct a relation
we must have ready all sets.
(ii) The pair primary key - foreign key, can find each other by
matching. However m-n relationship need some third agent, usually a
data entry person, who will pair every pair, it can be the millions of
these pairs. On the other hand if we have a function, then for every x
the function will compute the corresponding y.

3) Codomain
The functions don't have so strict codomain as a relation. The
codomain of function can be a superset of the range of the function.
For example Bourbaki aproach to the function is based on the set of
all functions from A to B. Here it is better to work with a codomain
than with the range of the functions.

This is longer text, but the function is really basic thing. In fact I
am surprise that people who are good at this field didn't write
anything.