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.