In Demon's third match he's wearing some sort of army pants. After winning his match he taunts Alex, while he's taunting him you can clearly see him wearing tight black shorts. Then as Demon leaves the canvas, he's wearing the same pants he wore during the match.
The sword that Alex steals at the beginning of the movie is curved with a tassel on the end. The sword that John sells back to him is a straight sword with a tassel on the end.
During the scene where Alex is to sell the katana in the hotel lobby, his partner says, "Stealing from James Leung is suicide." The character's name is David, not James.
Alex has the sword and throws it into Jon's car before getting placed under arrest. The police were close enough to Alex to see him doing this and none even bother to stop Jon from driving away.
The sword that Alex steals in the beginning of the movie is called a Japanese Samurai sword. It is in fact a Chines Jian, or double-edged straight sword.