You can't become "Invisible" on Facebook Chat. Instead you can appear "Offline" though still be on Facebook, just others cannot see that you are online.
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Incognito mode simply means there will be no record of your browsing or download histories. It will not make you 'invisible' to your Facebook friends.
Appearing offline is a key feature in a number of social and chat locations. At times, it would be improper to be involved in chats, so invisible mode is often employed. Invisible is one of the options under chat controls for Facebook mobile.
if you have a facebook message then it is already on facebook
use a proper facebook messenger, it's better plus has more features. use this http://www.facebook.com/apps/application.php?id=102332746051&sk=info you can be invisible while still seeing who is online. And be online to some friends, while appearing offline to others. Also has a cool away/busy function.
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