Continuity mistake: When Doug puts down the remote to grab Carrie's Walkman, the position of the remote changes between shots. (00:00:45)
Food Fight - S4-E13
Continuity mistake: When Arthur tells Doug to eat some salad he is either holding the bowl or not depending on the shot. (00:18:40)
Continuity mistake: When looking at the quote for the repair bill, its position on the table in relation to Kelly's menu is constantly changing. (00:08:40)
Continuity mistake: When Doug says he was reaching for a fig newton, the way Carrie is stood changes between shots. Note her right arm. (00:04:50)
Continuity mistake: When arguing about who is going to die first, the amount of whipped cream around Doug's mouth changes instantly. (00:19:20)
Continuity mistake: When talking to Deacon outside, Deacon has his arms folded or straight depending on the camera angle. This is before anyone comes out. (00:08:05)
Hi Def-Jam - S3-E13
Continuity mistake: When Doug is seeing the HD TV for the first time, he's holding his PDA in two hands which then becomes one hand when the camera has changed to a different angle. (00:02:20)
Continuity mistake: When Carrie gets her camera it changes position in her hands before she turns it around. Note the strap, it goes from left to right. (00:06:00)
Continuity mistake: Carrie asks Doug what he's doing when she walks into the bedroom. She then hangs her coat on a hanger and has the coat in her hands (as seen in her reflection in the mirror). Camera changes and she's shutting the wardrobe. (00:08:01)
Continuity mistake: When Carrie suggests about them looking after the kids to Doug, he has his arm up by his chest. When the camera changes it's lowered. (00:04:20)
Strike Too - S3-E6
Continuity mistake: When Stephanie is placing cones, she is looking at Doug with a concerned look upon her face. When the camera changes she's now facing away and laughing. (00:16:20)
Strike One - S3-E5
Continuity mistake: When Carrie says about smelling like the F train she has her hands together in front of her chest. When the camera changes angle, her hands are spread. (00:16:20)
Continuity mistake: When Doug is writing on a board, he holds the board at the end or the top depending on the shot. (00:03:30)
Continuity mistake: After Carrie says she's taking Doug's camera to her office party, the box magically appears on her lap. (00:09:50)
Continuity mistake: When talking in the bedroom, the magazines on the bed change from on the bed to on Carrie's ankles. (00:07:20)
Strike Out - S3-E7
Continuity mistake: The bottle of Lipton Ice Tea on the table rotates on its own without anyone touching it. (00:03:20)
Continuity mistake: Near the end when the Heffernans, Palmers, and "Shmenkmans" are having dinner and are in conversation, Carrie is shown sitting extremely close to Abby, who is on Carrie's left. But in shots in the middle of the conversation, there is significant empty space to Carrie's left.
Answer: To answer the question directly. Carrie and Doug do not have a biological child during the show's main run. In season 3 or 4 finale (cant remember which) Carrie has a miscarriage. Later on in the show they find out Carrie has 1 working ovary. Thus reducing the chance of them conceiving. By the final season, they couple agree to adopt from China and go to collect their child in the show's final episode. While on the flight back from China it's revealed that Carrie is once again pregnant. However the last scene of the show is a flash forward showing that Carrie goes full term with that child and they are struggling with both kids.
Ssiscool ★
Adding on to this answer, I read somewhere that producers were afraid of the show having a "before baby" feeling and then an "after baby" feeling. Some fans feel that main characters, or the show in general, decline in quality after a major change. For example, Jesse and Rebecca on "Full House" after they had twin sons, and Daphne on "Frasier" after she married Niles.