Continuity mistake: Closeau is at his desk preparing to light his pipe. The Pipe is in his mouth as he lights the match, the camera angle changes and now it is out of his mouth in his left hand as the camera zooms in, he puts it back in his mouth to light.
Continuity mistake: Closeau gets tripped by a man walking his dog while on the way home from the market. He starts spilling either salt or sugar all the way home. As he gets of the elevator at his home he leaves a trail out of the elevator and to his door. His neighbor lady comes into the hallway and he performs a little softshoe. Now the problem is, when he left the elevator he hadn't spilled much if any in there. When his neighbor makes her way towards the elevator, there is a big pile of the sugar/ or salt in the elevator that wasn't there just before the camera angle change.
Continuity mistake: An old man walks out from the Trattoria as the four cars go zipping back and forth by him. He walks out into the street then walks back to the front of the Trattoria and grabs a chair to sit and watch. The nitpick is the chair wasn't there when he came out of the Trattoria the first time, but it is when he goes back and sits down.
Continuity mistake: When the Godfather tells the girl at the side of the pool to take a swim, she jumps in with sunglasses on and comes back up without them. Later in the scene she asks the Godfather if she can get out and she has the sunglasses on again.