My customer asked me to develop a web app which include function chat with facebook but as I know, the Facebook Chat API and XMPP things has been deprecated since 04/30/2015.
But the function chat with facebook on Skype still working. Are there have any other approaches beside Facebook Chat API? I would like to hear an clear answer to explain for my customer.
No, there is no alternative to the Chat API. It has been removed and there is no way to implement chat in your App anymore. Skype may have a special deal with Facebook, or they are not upgraded to v2.0 yet - it´s a process that is still going on afaik.