if the mail function returns false, that means that there is no mail transfer agent set up with php.
in order for php's mail function to work, in the php.ini file, on unix servers the admin must specify the path to the mail sending program (sendmail, qmail....). php will return true if the mail successfully is piped to the mail agent (which doesnt necessarily mean the message gets sent) but it returns false if it doesnt even get to an agent.
it would appear it is their problem, and they need to go in the php.ini file and correctly fill in the sendmail_path value.
by default it ships like this in php:
; For Unix only. You may supply arguments as well (default: "sendmail -t -i").
;sendmail_path =
it should look like
sendmail_path = /usr/sbin/sendmail -t -i
its possible for your host to disbale phpinfo from working, as there are options in php.ini to disable certain functions.
if you have ssh access see if you are able to view /etc/php.ini (that is a common default location).
or write a php script to do this:
<?php
$out = cat /etc/php.ini;
echo $out;
?>
if permissions dont stop you from reading it, that should work.