Factual error: The main apartment in the show is located in Pasadena, Calif. When looking out the window in the background, mountains are shown in the distance and far away. In reality, Pasadena lies right underneath the long mountain range so the mountains would be up close, not far away.
Continuity mistake: At the end when Sheldon has got a burger, the amount of burger left changes depending on the camera angle. Most noticeable when he asks Penny about permanently reserving the table - between shots the lettuce disappears, the bun changes colour, and the bite marks change.
Character mistake: When Howard finds the stack of paychecks in Sheldon's drawer and wonders why Sheldon hasn't cashed them, Sheldon explains that he's saving them to purchase items that have yet to be invented and he doesn't trust banks. But just keeping them in his drawer doesn't do anything, because checks are voided after a set period of time and they would eventually become worthless.
Continuity mistake: The Stevenson Award in Sheldon's office reads "Sheldon Cooper, PhD." But, in the "Dennis Kim" episode we are told Sheldon was 14.5 years old when he earned that award - 1.5 years before earning his PhD at age 16, so the award should not show the title of PhD.
Continuity mistake: After the 'Ramona' conversation between Sheldon and Penny ends, Sheldon leaves Penny alone on the couch. Penny looks up to the "Amy and Penny" painting and says "Don't look at me like that, I tried." In S10 E10, Amy gave the painting back to Penny, which was hung in Leonard's living room.
Revealing mistake: When Sheldon goes to Penny's apartment and they do a scene study together, they both read lines from a multi-page script that Sheldon brings with him. On a couple of occasions the camera cuts to a side view of Sheldon and we get to see the page of the script that he's currently reading from. The pages are completely blank.
Character mistake: When Penny says she smells Chinese food and Sheldon corrects her by telling her she's slipping because it's Thai, Sheldon is eating the food with chopsticks, even though in a previous episode he uses a fork and spoon because that's how Thai people eat their food. (00:19:15)
Character mistake: When Sheldon arrives at the apartment with his Nintendo 64 in the box, he tells Leonard that they can play Super Mario 64. He then proceeds to say, upon finding a memory card, that they can pick up where he left off. Super Mario 64 does not make use of the memory card for progression, all of that data was saved on the cartridge. (00:07:25)
Continuity mistake: In the laundry room when Penny is talking to Sheldon, Sheldon takes some time arranging a blue shirt on his plastic laundry folder. Then it suddenly turns into a pair of socks, which he folds and then picks up the blue shirt again. There was not enough time for him to switch out the shirt for the socks. (00:06:30)