Continuity mistake: After Phileas and Passepartout escape the art gallery we see them come up through a man-hole cover in the street. We see the girl exit the sewer and then Phileas. Just as he is getting out there is a couple walking past. The camera changes and they have disappeared, and we then see them walk past again.
Continuity mistake: When Passepartout and Fogg come out in their car, with a big propeller in front of it, an inspector stops them. The inspector then pulls the handle which makes the car start off. A strap then pulls the inspector over and you see the strap barely touching his heels. In the next shot you see the strap twined around the inspector's ankle as he is being dragged after the car.
Revealing mistake: When Fogg pulls the knife out of Passportout's buttock you can see that the knife is attached to something under his pants.
Plot hole: The hot air balloon carries Fogg and friends quite a distance, but where is the heat source?
Deliberate mistake: If Fogg and company are well on their way around the world, how on earth did Inspector Fix beat them to India?
Continuity mistake: When Fogg meets the character played by Rob Schneider, Rob says "Now, your stink is your most powerful weapon..." while pointing with his fingers. When it cuts to Fogg you see Rob's hands suddenly appears on Fogg's shoulders.
Continuity mistake: When Passpartout is making hot tea, he throws the water into Fix's crotch. A few minutes later, when Fix is looking back at Passpartout, his pants are fully illuminated and are dry.
Answer: In order to stop Fogg from completing the journey, his rivals at the Academy wired (telegraphed) details to confederates in the countries he visited with instructions to waylay him.