When Beckett is reading Heat Wave in the bathroom, her finger positioning changes between shots.
When Beckett is tapping on the window to alert Sue that the con was figured out, her finger positioning on her badge changes from shot to shot.
Castle refers to Gray as "agent Gray" of the CIA; Gray would be an officer of the CIA; (Intelligence officers would recruit agents in the field.)
When Castle catches Beckett in the bathroom stall reading the book, you can see that the pages are blank.
When Esposito receives the calls about Fletcher using all his identities to reserve rental cars and flights to non-extradition countries, he is holding the cellular phone upside down.