Continuity mistake: When we see the professor's clone inside the tube there is a panel attached to a metal post going down to the bottom of the tube. When the professor hits the button to empty the tube of fluid it cuts to a close-up of the clone's feet and the metal post is suddenly gone.
Continuity mistake: When Professor Farnsworth places the envelope containing his holo-message on the table, we get a quick glimpse of the front of the envelope, and it is blank, without any writing on it. He then walks away with the Sunset Squad robot, without taking time to address the envelope. When the crew finds the message, the words "For my crew" appear on the envelope.
Continuity mistake: When Bender makes the joke about where the professor would be without all the students who love and respect him, we see the professor sitting with nothing in front of him on the table. When it cuts, two glasses and a blue cup suddenly appear.