I'm including this library "ESP8266WiFi.h" at the top of my sketch. I can find documentation on WiFiServer.available(), but didn't find for WiFi.connected(), which is called in a server sketch after getting the client.

Clearly it's a test whether (once? I believe) the client is connected or not (and hasn't been explicitly closed in the server's side). It may seem obvious but some documented info would be helpful though.