No Orleans - S4-E16
Continuity mistake: When Spence is seen eating a snow cone for the first time, the level of it changes between shots. (00:06:10)
Continuity mistake: Arthur gives Holly a book at the start and she has it in her right hand. Camera cuts and now it is in her left hand. (00:03:20)
Continuity mistake: In the establishing shot of the gold course, it's a bright and sunny day. A few moments later when they are all teeing off it's drizzling rain and you can see Doug's breath. (00:09:35 - 00:11:30)
Continuity mistake: When Carrie takes a drink from her cup of tea she is holding her cup with 1 hand in one shot and 2 in a second shot. (00:14:30)
Continuity mistake: When Carrie is talking to Tina about playing 3 holes, her hands are outstretched. Camera cuts and her gloved hand is now on her hip. (00:16:40)
Character mistake: When Carrie finds Tina, she mentions the "stuff" that Tina listens to at work, and Tina tells her that it's salsa. We know from previous episodes that Carrie likes salsa dancing, so shouldn't she know salsa music?
Continuity mistake: When Carrie finds Doug in the kitchen in the middle of the night, her arms are crossed with her hands tucked into the sleeves. After she asks him about his pay check from IPS, the shot changes and her hands are now on her hips. (00:10:50)
Continuity mistake: When Doug is talking to Carrie in the kitchen about his ideas for the sandwich shop, lights reflecting in the living room windows behind Doug turn off and then back on when the shot changes between Doug and Carrie. (00:11:00 - 00:11:35)
Revealing mistake: When Doug's mum is writing down the note about the T-11's, the writing is already on the pad. (00:14:40)
Continuity mistake: While at Doug's high school reunion, he and Carrie are dancing. Carrie has her purse on her shoulder. When they go for a beverage, the purse is gone. She was able to put her bun back without having a purse swinging in her face.
Two Thirty - S4-E24
Plot hole: In this episode Arthur scams Deacon and several other parents for phony piano lessons because he doesn't actually know how to play the piano. However, earlier in the series (Art House episode, S1 - E24) we learn that Arthur can play the piano when he temporarily moves out and after his house warming party Doug says to Carrie, "Who knew Arthur could play the piano like that?"
Shrink Wrap - S4-E25
Continuity mistake: In the scene when the therapist asks Arthur to recall a specific incident about not being heard from his childhood, he is playing with his pen cap. The shot cuts and he is instantly holding his note pad with both hands. (01:35:40)
Answer: Sara hasn't been mentioned in later episodes, and it suggested in later episodes that she never existed, and that Carrie is an only child. Kevin James stated that the producers couldn't think of an good plots to use with Sara, so they left her out.