Pues eso, inicialízala antes:
$result="";
if(!$_POST['action']){
//We are redirecting people to our shoutbox page if they try to enter in our shoutbox.php
header ("Location: index.html");
}
else{
$link = connect(HOST, USER);
switch($_POST['action']){
case "update":
$res = getContent($link, 20);
while($row = mysql_fetch_array($res)){
$result .= "<li><strong>".$row['user']."</strong><img src=\"css/images/bullet.gif\" alt=\"-\" />".$row['message']." <span>".$row['date']."</span></li>";
}
echo $result;
break;
case "insert":
echo insertMessage($_POST['nick'], $_POST['message']);
break;
}
mysql_close($link);
}