Are PostgreSQL functions transactional?

PostgreSQL 12 update: there is limited support for top-level PROCEDUREs that can do transaction control. You still cannot manage transactions in regular SQL-callable functions, so the below remains true except when using the new top-level procedures.

Functions are part of the transaction they're called from. Their effects are rolled back if the transaction rolls back. Their work commits if the transaction commits. Any BEGIN ... EXCEPT blocks within the function operate like (and under the hood use) savepoints like the SAVEPOINT and ROLLBACK TO SAVEPOINT SQL statements.

The function either succeeds in its entirety or fails in its entirety, barring BEGIN ... EXCEPT error handling. If an error is raised within the function and not handled, the transaction calling the function is aborted. Aborted transactions cannot commit, and if they try to commit the COMMIT is treated as ROLLBACK, same as for any other transaction in error. Observe:

regress=# BEGIN;
BEGIN
regress=# SELECT 1/0;
ERROR:  division by zero
regress=# COMMIT;
ROLLBACK

See how the transaction, which is in the error state due to the zero division, rolls back on COMMIT?

If you call a function without an explicit surounding transaction the rules are exactly the same as for any other Pg statement:

BEGIN;
SELECT refresh_materialized_view(name);
COMMIT;

(where COMMIT will fail if the SELECT raised an error).

PostgreSQL does not (yet) support autonomous transactions in functions, where the procedure/function could commit/rollback independently of the calling transaction. This can be simulated using a new session via dblink.

BUT, things that aren't transactional or are imperfectly transactional exist in PostgreSQL. If it has non-transactional behaviour in a normal BEGIN; do stuff; COMMIT; block, it has non-transactional behaviour in a function too. For example, nextval and setval, TRUNCATE, etc.

As my knowledge of PostgreSQL is less deeper than Craig Ringer´s I will try to give a shorter answer: Yes.

If you execute a function that has an error in it, none of the steps will impact in the database.

Also, if you execute a query in PgAdmin the same happen.

For example, if you execute in a query:

update your_table yt set column1 = 10 where yt.id=20;

select anything_that_do_not_exists;

The update in the row, id = 20 of your_table will not be saved in the database.

UPDATE Sep - 2018

To clarify the concept I have made a little example with non-transactional function nextval.

First, let´s create a sequence:

create sequence test_sequence start 100;

Then, let´s execute:

update your_table yt set column1 = 10 where yt.id=20; select nextval('test_sequence'); select anything_that_do_not_exists;

Now, if we open another query and execute

select nextval('test_sequence');

We will get 101 because the first value (100) was used in the latter query (that is because the sequences are not transactional) although the update was not committed.

https://www.postgresql.org/docs/current/static/plpgsql-structure.html

It is important not to confuse the use of BEGIN/END for grouping statements in PL/pgSQL with the similarly-named SQL commands for transaction control. PL/pgSQL's BEGIN/END are only for grouping; they do not start or end a transaction. Functions and trigger procedures are always executed within a transaction established by an outer query — they cannot start or commit that transaction, since there would be no context for them to execute in. However, a block containing an EXCEPTION clause effectively forms a subtransaction that can be rolled back without affecting the outer transaction. For more about that see Section 39.6.6.