Continuity mistake: The President of the US is watching football on channel 1. Dreyfus interrupts the broadcast and an aide flips through the channels to discover that Dreyfus is on all of them - the TV ends up on channel 9. Throughout the rest of the scene, the TV switches back and forth between channels 1 and 9 without human intervention.
Revealing mistake: When Clouseau is on the parallel bars, note the switch as he jumps and turns from Peter Sellers to stuntman, then note again as the stuntman falls down the stairs; the stuntman finishes falling behind the right hand side of the settee, and up pops Peter Sellers on the left hand side of the settee giving the apparant impression he has just fallen down the stairs himself.
Continuity mistake: When Clouseau is on the front page of the newspaper after being arrested in the Soho club the date on the front of the newspaper is Monday July 12th, but two other events in this film are in October, the beer festival and the disappearance of the UN building. The events of the film don't cover that long a timespan.