Continuity mistake: When Prot takes his sunglasses off for his "return" (because he would supposedly no longer need them due to the lower light levels on K-Pax), he places them on the nightstand upside-down, with the earpiece flush to the table. Two subsequent shots (including an elaborate effects shot) show the sunglasses right-side-up. (01:48:45 - 01:53:30)
Continuity mistake: When the woman who never leaves her room is seated at the table with some other patients, she's talking to the doctor and closes her hand fan. The very next scene shows the fan open again.