If isset $_POST

Use !empty instead of isset. isset return true for $_POST because $_POST array is superglobal and always exists (set).

Or better use $_SERVER['REQUEST_METHOD'] == 'POST'

If you send the form empty, $_POST['mail'] will still be sent, but the value is empty. To check if the field is empty you need to check

if(isset($_POST["mail"]) && trim($_POST["mail"]) != "") { .. }

From php.net, isset

Returns TRUE if var exists and has value other than NULL, FALSE otherwise.

empty space is considered as set. You need to use empty() for checking all null options.

Most form inputs are always set, even if not filled up, so you must check for the emptiness too.

Since !empty() is already checks for both, you can use this:

if (!empty($_POST["mail"])) {
    echo "Yes, mail is set";    
} else {  
    echo "No, mail is not set";
}