Continuity: 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).

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: 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.