I wrote this post because I ran across this issue when debugging why tables with triggers/functions that were not getting replicated to slaves. The problem appears when binlog_format is set up as STATEMENT. You can check the ‘change log’ checking the following link for more information about default values .
If you have non-deterministic functions  that insert or modify your data with dynamic SQL like this :
CREATE FUNCTION `insert_dynamic_data` (in_param VARCHAR(30)) RETURNS bigint(20)
INSERT INTO `table_to_ins` (param) VALUES(in_param);
/* You can also add here @@warning_count or row_count() check
and return -1 to catch errors*/
The problem starts when you plan to user “replication” through binary logs events. In that cases, you must consider:
“If a stored program that modifies data is nondeterministic, it is not repeatable. This can result in different data on a master and slave, or cause restored data to differ from the original data.” 
When you create a function, by default is NOT DETERMINISTIC, which means that for each execution could the code will be executed again. In the other hand, DETERMINISTIC will check if the parameters are the same and if is that condition comes true, will return the same result as before without executing again the code.
If you want to create the function with the binlog_format in STATEMENT and log_bin activated without the log_bin_trust_function_creators, you will get the following error:
mysql> DELIMITER $$
mysql> CREATE FUNCTION fx_pru () RETURNS int
-> insert into prueba select ‘po’,round(rand()*100), rand();
-> RETURN 1;
ERROR 1418 (HY000): This function has none of DETERMINISTIC, NO SQL, or READS SQL DATA in its declaration and binary logging is enabled (you *might* want to use the less safe log_bin_trust_function_creators variable)
What happened so? The problem was that the funciton were created previous to activate the log_bin.
You have to ways to fix it: activating binlog_format as ROW or declare in your /etc/my.cnf the log-bin-trust-function-creators which allows insertions in the binary log without enforcing deterministic property. In the specified case, we cannot declare this function as DETERMINISTIC because it uses dynamic values in the parameters (always we expect different values and different results).
What happens with the execution of the function? Basically, it creates a lock contention ( if you usually use Nagios, you will see a mysql-lock-contention alarm), but MySQL will not raise any error or alarm, which causes confusion when trying to find the error. Other statements that run outside the function will be executed normally. In the previous example, this function was only for inserts, but every statement inside the function will not work. Derived from this problem, statements inside the function will not be executed directly.
You wouldn’t see any errors. The only thing you could realize is the lock contention. The lock is generated because the statement hangs a lock into the table trying to reach a commit that never comes.
It is a bug? No, is a feature. It prevents unknown and potentially slow executions from being replicated, causing lagging into the replication process.