At the beginning, after Penny shoots Leonard with a toy rocket, the launcher on the table is suddenly aimed back at Penny instead of where Leonard is still standing.
When Sheldon sings "Soft Kitty" to the Professor, he smiles. It then cuts to a side view of the Professor and he is not smiling.
After having to walk up 3 flights to pick up Dr. Jeffries, the paramedics should have reported the landlord to the proper authorities to get the elevator fixed, but that would have changed the makeup of the show.
When Raj arrives over to Howard and Bernadette's. Howard leaves the door ajar, when the camera focuses on Raj, the door is shut properly.
Sheldon repeatedly addresses Professor Proton as Mr. Jeffries, even after Arthur Jeffries has told him that he has a Ph.D. As a stickler for detail, and as a mark of the respect he obviously feels, Sheldon should address him as Dr. Jeffries.