Frozen Pop - S2-E15
Continuity mistake: When Carrie is telling Arthur to move upstairs, the blanket Arthur has got changes position instantly. (00:02:40)
Frozen Pop - S2-E15
Continuity mistake: When Carrie is talking to Arthur at the start, Arthur's hands are in his lap in one shot and clasped together in the next. (00:02:45)
Frozen Pop - S2-E15
Continuity mistake: When Doug goes back into the house at the start he opens the door with his right hand. Camera cuts to inside the house and now it's his left hand on the door handle. (00:01:35)
Party Favor - S2-E13
Continuity mistake: When Doug is mimicking the Japanese chef, the towel on the side moves positions without being touched. (00:01:00)
Party Favor - S2-E13
Continuity mistake: At the start, Doug has a spatula in one hand and a whisk in the other. Camera does an instant cut and they have swapped hands. This is not to be confused with when he is messing about and you see him swap the items between hands. (00:00:50)
Continuity mistake: When talking to Doug in the back yard, Carries hands are changing position between instant cuts. In some shots they're together in her lap in others she's holding Doug. (00:04:00)
Time Share - S1-E22
Continuity mistake: When Doug Carrie and Ritchie are in the kitchen discussing how long Ritchie has to keep seeing Dorothy, Doug is making a snack involving some lettuce. The size, shape and position of this lettuce is constantly changing between subsequent shots. (00:15:20)
Hungry Man - S1-E21
Continuity mistake: At the very end when Carrie walks out the level of foam on Doug's beer bottle changes between shots. (00:21:20)
Continuity mistake: At the start when Arthur is ranting to Doug, the way Doug holds his coffee flask differs ever so slightly between shots. (00:07:00)
Court Date - S1-E17
Continuity mistake: When Doug, Carrie and Ritchie are talking about the cop having not called her yet, Doug's hands are either on or off the bottle of beer, and the level of beer/foam changes between shots too. (00:16:30)
Court Date - S1-E17
Continuity mistake: When Arthur storms out saying that was his drug speech, Carrie has her wrists in the table in one shot and under the table in the next. (00:15:10)
Continuity mistake: Doug comes out from the payphone and asks Carrie where did everyone go. When Carrie replies, the way she holds her glass changes between shots. (00:17:50)
Fixer Upper - S1-E12
Continuity mistake: When Doug is talking to Spencer in the subway, the people in the line behind him change instantly from 2 men to 1 man and 1 woman. (00:19:30)
Continuity mistake: When making a PB&J sandwich, the jar of peanut butter rotates by itself. Note the label. (00:09:30)
Visible crew/equipment: When the store clerk walks off from talking to Arthur an orange T marker is visible where he was stood. (00:14:50)
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.