The Cooper-Hofstadter Polarization - S1-E9
Continuity mistake: The video of the fight we see on Youtube is different to what we saw earlier in the episode.
Continuity mistake: In nearly every episode, Sheldon demonstrates his OCD when he knocks three times on a door, says a person's name, and repeats that sequence twice, for a total of three iterations, but in this episode he radically departs from that behavior not once but twice. First, wanting to talk to Leonard, he knocks four times, says "Leonard" and repeats the sequence three times for a total of four iterations. Later, wanting to talk to Penny, he knocks four times and says her name, repeating that sequence at least four more times. Given that this was the first season, I assume the writers had not yet settled upon the 3x3 routine as a comic behavior.
The Grasshopper Experiment - S1-E8
Continuity mistake: When Raj is drinking with Lalita in the bar Penny works at, in one shot he raises his Grasshopper drink to his lips with his left hand. In the next shot it's in his right (this is just before he says "Oh my God, you've lost so much weight").
Character mistake: When Leonard and Sheldon sit down for lunch at work, Sheldon grabs the salt and pepper and begins to shake the "shakers" over his food. However, these are grinders and not shakers. He would have had to twist the tops in order to get any salt or pepper. (00:16:10)
The Pancake Batter Anomaly - S1-E11
Continuity mistake: The way Penny holds her notepad when talking to Sheldon at the Cheesecake factory keeps changing instantly. (00:10:10)
Continuity mistake: In "The Luminous Fish Effect" (S1E04) Sheldon tells Penny he weighs 140 pounds. In "The Porkchop Indeterminacy" (S1E16), he tells his sister he weighs 165. 25lbs difference would be a noticeable change, but he looks the same.
Suggested correction: Lying to Penny but not to his sister, not inconsistent with the characters and his relationships.
The Pancake Batter Anomaly - S1-E11
Continuity mistake: After Leonard gives Penny her pile of mail, she alternates from holding the pile by her side to at her front between each shot. (00:01:10)
Continuity mistake: When Penny asks Sheldon "What trauma?" in the electronics store, Sheldon holds two boxes - one with an orange side on top of one with a black side. In the next shot the black sided box is on top of the orange one. The boxes then continue to swap places during the whole scene. (00:11:50)
Character mistake: Raj cannot talk with a woman around, but when Leonard claims they'd have to cut him in half to play teams, he blatantly argues back, and Penny is present with them. Regardless of if he was facing her or not, he shouldn't have said anything.
Continuity mistake: During the whole opening scene, the way in which Leonard holds the letters in his hands keeps differing.
The Pork Chop Indeterminacy - S1-E15
Continuity mistake: When Sheldon says that his sister was sent to spy on him, his arms are behind his back. Camera cuts and they are now at his side.
Continuity mistake: When Leonard and Sheldon are talking with Penny during the scene where she is being allowed to sleep on their couch, the pullstring on her hooded jacket keeps alternating from hanging loose to being caught inside the top of the jacket. (00:07:40)
The Big Bran Hypothesis - S1-E2
Audio problem: When Penny is talking to Raj on the stairs to the apartments, during some shots we see her from behind. In one shot, we hear her say something like "And I'm freaking out", but you can tell by the way her jaw moves that she was actually saying something different. (00:14:50)
The Pancake Batter Anomaly - S1-E11
Continuity mistake: When Sheldon walks out his bedroom wrapped in a comforter, he drops it on the higher level before the drop down to the kitchen. In the next shot as he walks away from the fridge, half of the comforter is now spread over the lower level without anyone moving it. (00:05:10)
The Middle Earth Paradigm - S1-E6
Continuity mistake: When Leonard is talking to Sheldon in the kitchen about the party, Sheldon is pouring juice. In one shot he places the bottle cap on the bottle yet in the next shot his hand is down on the counter with the lid in it. (00:04:50)
The Nerdvana Annihilation - S1-E14
Continuity mistake: In the scene where Leonard is packing up his collectables to sell, at 15:32 he is dusting what appears to be a Hasbro Marvel Legends Yellow jacket figure, which also has a yellow post-it on the back. At 15:42 he is still dusting a figure but he now holds a different (darker but not yellow) figure from the same series, with no post-it on the back, and the Yellow jacket figure is now on the table (house left of the box of collectables). At 15:56 The Yellow jacket figure has disappeared from the table, and it is not the figure in Leonard's hands either. (00:15:30)
Continuity mistake: When Sheldon is talking with Leonard in the kitchen at night, as he pours himself a mug of milk the mug moves between shots. (00:04:50)
The Hamburger Postulate - S1-E5
Continuity mistake: Leonard and Leslie are rehearsing the violin and cello in the guys' living room, and when they leave the room to go to Leonard's room, Leonard's cello and stand are still in front of the comfy chair. When Sheldon gets home, the cello and stand are gone, and only Leslie's closed violin case is left on the sofa. And she picks it up and leaves with it. How did the violin get back in the case.
Continuity mistake: Sheldon puts the bag of takeout into the lobby trash can and as he does so, the whole bag turns sideways. At the end of the scene, he takes the bag out of the can and it's upright without him turning it.
