Plot hole: In this episode Arthur scams Deacon and several other parents for phony piano lessons because he doesn't actually know how to play the piano. However, earlier in the series (Art House episode, S1 - E24) we learn that Arthur can play the piano when he temporarily moves out and after his house warming party Doug says to Carrie, "Who knew Arthur could play the piano like that?"
Continuity mistake: In the scene where Arthur is on the phone with PBS to complain about getting a "Civil War Series" box set, the actual set of tapes changes drastically. When it's on the table and when Carrie brings it over to the counter its a pretty large, has an American flag motif, and says "Civil War" across the top. When it cuts back to it sitting on the counter its a much smaller box with a plain design and says "The Great War" along the bottom. It's obviously a different set, not just the camera angle. (00:14:40)
Revealing mistake: In the final scene, Carrie starts to kiss Doug and pushes him onto the bed. The camera angle switches from what would be outside the window to beside the bed. The scene was done in one take from 3 camera angles. As the camera pans left you can see the center window is completely open because they had just pulled the camera out of it and didn't get it closed yet, and you can watch the window be closed by an invisible hand from underneath. (00:28:00)