Continuity mistake: When Carrie is reading from the leaflet, the way she holds it changes depending on the shot. (00:17:35)
Visible crew/equipment: When Doug is lying on the floor with the phone on him, the camera zooms in and you can see a boom arm reflected in the phone. (00:19:35)
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)
Character mistake: In this season we see Doug smoking cigars and cigarettes. However in S2 E1 Queasy Rider, Doug is shown to absolutely detest the smell and taste of smoke from these. It is highly unlikely that he would then take up smoking.
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)
Strike One - S3-E5
Visible crew/equipment: When Doug is listing all the extras for the truck, you can see tape markers on the floor behind them. Specifically, one in between the feet of an extra that is standing behind them. (00:08:50)
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)
Visible crew/equipment: When Doug is outside talking to Deacon and Arthur opens the kitchen door, as the door swings back you can see all the set lights reflected in the glass. (00:08:20)
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.