Continuity mistake: In the scene where Carrie and Doug are in the kitchen and Carrie is explaining why she wants to quit school, she fills a glass of water to the top of the glass. A few shots later, she sets the glass down on the counter without taking a drink and it is less than half full.
Continuity mistake: When Doug is ringing Arthur at the beginning of the episode, there are 2 lunch boxes on top of the lockers behind Doug. During different shots 1 disappears and reappears. (00:00:45)
Continuity mistake: At the start there are 2 letters on the table in front of Arthur. When Doug rings for the 3rd time they have disappeared. (00:00:50)
Continuity mistake: Right at the start when Carrie gives a sigh, she shuts the menu. In the next shot it's back open. (00:00:25)
Continuity mistake: At the dinner table while talking about being obscure at work, Doug starts pouring his beer into a glass. But immediately in the next shot he is finished pouring and both bottle and glass are on the table. A moment later he picks up the beer and pours it into the glass a second time. (00:03:45)
Continuity mistake: At the beginning Carrie is dishing out potatoes from a bowl with a large black spoon, and as the camera changes angles her arm has not moved and she is now holding a small fork getting chicken from a different bowl. (00:01:40)
Continuity mistake: When Carrie is reading from the leaflet, the way she holds it changes depending on the shot. (00:17:35)
Continuity mistake: When Doug says to Carrie that he doesn't even know her anymore, Carrie's laptop and books are open over the desk. However when the camera cuts to her standing up, she has got the laptop folded up and her books in her arms instantly. (00:10:35)
Continuity mistake: At the start of the episode when Arthur is on about kissing ass, the way his napkin is positioned constantly changes. (00:03:45)
Continuity mistake: Doug sits down near the start after getting a drink and proceeds to pour a drink. When it cuts he has instantly stopped pouring the drink. (00:03:40)
Continuity mistake: At the start when Arthur is on about Al Gore, he has chicken in his hands, but when the camera cuts, he now has a fork and napkin. (00:01:35)
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)
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.