Final Exam - S1-E1
Deliberate mistake: Cyborg, Starfire, Raven and Beast Boy are kicked out of the Tower by the HIVE, and the trio watches them from the Operations/Main Living room panorama window. But whenever Jinx, Gizmo and Mammoth are seen (before they send in the tidal wave), the same window is on their right-hand side, whereas there should be a solid wall separating Ops from the side wings.
Final Exam - S1-E1
Continuity mistake: After the Titans are evicted from their home and grieve over their loss, before Robin re-enters, as the Titans are gazing at the ocean, Cyborg's arms are attached to him, despite having lost his right one, which is still in Gizmo's possession.
Chosen answer: It is located in California, as shown in "Teen Titans: Trouble in Tokyo". Possibly San Francisco as seen in the comics.
MasterOfAll