Continuity mistake: After Cliff talks to Peter, Theo is told to run to the neighbors across the street. However, he can be seen in the background of some shots when the twins do their game. As soon as the twins finish, Theo is shown coming back into the house.
Other mistake: It is daylight outside when Cliff and Clair are talking, but when he turns the lights off to try on his light-up Father's Day gifts, it gets pitch black in the room.
Deliberate mistake: In the very first episode, the kids are fighting and arguing all throughout the house. Claire then asks Cliff, "Why did we have 4 kids?", referring to Denise, Theo, Rudy, and Vanessa. In all of the following episodes, they actually have 5 kids. I guess they decided to add Sandra (the oldest Huxtable) later down the road.
Continuity mistake: The father was known as Clifford in the beginning but the name was changed to Heathcliff (you can see this on the nameplate for his office).
Back to the Track, Jack - S1-E20
Character mistake: When the race ends and the kids are cheering, you can hear Denise call Cliff "Bill" instead of "Dad".
Mr. Fish (a.k.a. Goodbye, Mr. Fish) - S1-E2
Continuity mistake: While some of the family is eating, the cup in front of Denise disappears than reappears.
Other mistake: During the scene where the twin sisters were singing their song, Theo was asked by Cliff to go to Peter's house and ask his mother if he could stay. Theo left immediately after this. But if you look closely while the twins are singing, you'll see Theo's arm in the scene. When the twins finish performing, they sit down. After this, you see Theo coming through the door after having just gone to Peter's house.
Breaking With Tradition - S1-E6
Continuity mistake: Theo and Cliff walk into the kitchen and Cliff sets the basketball on the counter. Theo walks upstairs. When the camera is back on the counter the basketball is gone.
Continuity mistake: After Theo gives up an extra $100 for clothes, he is holding $300 left in his hand. However when the camera switches back to Cliff, who asks how much is left, it switches back to Theo, who is only holding $200 left.
Continuity mistake: At the very end of the episode Clair's glass is suddenly fuller after Denise gets done singing.
Continuity mistake: When Cliff is giving the kids their dinner, the pan cover keeps changing position.
Continuity mistake: The amount of water in Clair's cup changes a couple of times, most noticeably after Cliff tells her what she is eating, she takes a drink and the cup is about 1/3 full. She gets up to go find Theo, and now there is very little water in the glass.
Rudy's Sick - S1-E12
Continuity mistake: When Rudy is lying in her bed when Cliff is telling her about the germs in her body, her position on the bed changes between shots.
Mr. Fish (a.k.a. Goodbye, Mr. Fish) - S1-E2
Continuity mistake: During the scene when Cliff is explaining to Rudy that her fish is dead, the fish food disappears at one point then reappears.
Chosen answer: That is the song "Open the Door Richard."