Continuity mistake: Travis walks in the building where he is greeted by the 'timekeeper' of the prostitute. He asks for money for the room. Notice he has just a toothpick in hand, but when the shot changes and he checks his watch, he is holding a gun. (01:19:00)
Continuity mistake: Iris walks into the room after Travis talked with the timekeeper. She moves past the beaded curtains and De Niro follows her. The bead strings oscillate with a difference of pace/momentum depending on the angle (they move more in the angle with camera behind Jodie Foster, because she really passes through them, and they are moving less in the shots with De Niro's face). (01:19:25)
Continuity mistake: In the hotel room, Iris is taking her shoes off. The heart pillow next to her on the couch changes position depending on the camera angle. (01:20:10)
Continuity mistake: When Iris and Travis are having breakfast together, at the beginning of the scene, she is wearing green sunglasses. She eventually takes them off, but when she puts the sunglasses back on, the sunglasses are now purple with a gold rim. (01:25:00 - 01:29:00)
Continuity mistake: During the breakfast scene, Iris has her hair suddenly underneath the shades' arms, just for a single shot. It's the one when he asks her "You call that being hip?" and when asks who's a killer. (01:25:35)
Continuity mistake: When Iris sobbing says that Sport never killed anyone, her carb-bomb made of bread, jam and pure sugar is looking quite messy, with the slices of toast not quite matching in rotation and raised. When she picks them back up from the plate though, they are all perfectly matching and pressed tightly together. (01:26:00)
Continuity mistake: During the breakfast scene Travis calls Sport "a dope shooter." He puts his fork down saying that line (you can hear the noise too) but with the camera back on Jodie Faster, Robert De Niro's hand is at the border of the frame holding the utensil exactly like before. (01:26:20)
Continuity mistake: The breakfast scene ends with Jodie Foster having a bunch more toast on her plate than at the start. (01:26:40 - 01:29:20)
Continuity mistake: While De Niro is talking to Jodie Foster you can see a bus out the window. When the screen switches to Jodie Foster's view you can see the same bus driving, but then it changes back to De Niro's view you see the same bus driving up to the window again. This keeps happening throughout the scene, the bus keeps driving around the building. (01:27:00)
Continuity mistake: During breakfast, Iris suggests Travis that he could come to the commune in Vermont with her. De Niro is making a joker-esque big grinning smile in the two following close-ups, but there's another shot in between, with Robert De Niro visible only from the side in foreground, and he is not smiling at all in that. (01:28:10)
Continuity mistake: Travis Bickle is preparing for his mission. He puts the flowers in the sink and burns them, he cuts one sleeve from his shirt so he can accommodate a special retractable holster, and then...then he is writing a note for Iris, wearing a shirt with both sleeves, in a room still full of wilted flowers. (01:33:00)
Other mistake: When Travis sends money to Iris, he addresses the envelope to 240 East 13th st. At the entrance to the building Iris lives in (visible after Sport's corpse is shown), the street number is 226. (01:33:40 - 01:44:50)
Visible crew/equipment: Travis Bickle got all geared up to go kill the senator. There's an opening panning shot of Charles Palantine arriving at the location. As the relatively small scope of the meeting implies, but also as shown in the rest of the scene, there's no camera crew following Palantine, no big crane rig or anything. Yet you can easily spot people in this crowd who don't care about the Senator but look at the movie camera. One of the followers is particularly hilarious; on the left of the frame, he keeps waving his hand at the camera, tries to follow Palantine and direct the gesture at him, and then goes back to signaling the camera. (01:33:50)
Revealing mistake: When the Senator finishes his speech and steps off the stage, Travis begins to move making his way through the crowd. Several extras are recycled appearing in inconsistent positions - look no further than the black woman with a bright flower patterned pink shirt; she is simultaneously in front of Travis as he begins to move in the outer section, and right between Charles Palantine and the stage in the shot immediately after. And then she keeps popping up where she's not supposed to be, continuously. (01:36:20)
Other mistake: Travis built a special arm holster for a gun that allows him to pull a gun easily, popping it in hand while his arm is already outstretched. It's the weapon he prepared for in the iconic "You talkin' to me?" scene, and practices handshaking with it. We see a scene when he prepares to go kill the Senator cutting the sleeve of the shirt off. And yet, when the actual attempted hit is shown, Travis has his hand (the gun hand) inside his jacket, alerting security. (01:36:30)
Visible crew/equipment: When Palantine is walking through the crowd, several extra don't look at all where he is supposed to be, but at the camera, either staring directly at it breaking 4th wall when it's in De Niro's close-ups, or even more blatantly turning around (and away from the Senator) to look up where the camera crane is. (01:36:35)
Other mistake: The man (turns out to be a mafia bigshot) who goes to Iris' pimp Sport asks him if she's in the usual building, and walks off, no doubt about his intentions. It's broad daylight, and it's summer. Yet Travis, who is also on booze and pills after failing to kill the senator and surely did not bid his time, arrives in East 13th Street only when it's already pitch black outside, and the john is still with Iris in the 'work' room, with them being fully clothed and with the bed made (but she's on her knees so they are 'just about to do it'). (01:37:30 - 01:39:45)
Visible crew/equipment: In the scene where Robert De Niro shoots Sport the pimp in the stomach, the bullet supposedly goes through Sport's back, and blood splatters onto the doorway framework behind him. However, as Sport falls to the ground, look closely, and you can see a wire dangling from his lower back, which was used by the special effects team to detonate the fake blood packet. (01:39:05)
Revealing mistake: Sport initially is cautious enough to inquire Travis about a gun, and suspect him of being a policeman. Forgetting the fact that he's suddenly dumb enough to not recognize just a few days later the weirdo with a huge mole on the cheek he talked to, just because he shaved his hair (maybe he was stoned), he still is approached by a guy with a hand in his pocket, in a dark street, making weird questions, who asks him if he has a GUN, and that then says "Suck on this." That already should make him at least quite jumpy, but then, at this point Robert De Niro pulls the gun out of his pocket...but flubs it, it does not happen smoothly. For a good couple of seconds Harvey Keitel stands in front of him motionless, unfazed like no human being would, just because there is no cue for the actor to do anything but stand there like an idiot and get shot. (01:39:10)
Continuity mistake: When Robert De Niro shoots the pimp's hand off in the hallway (shortly before shooting Sport the second time), blood splatters on his face, across his left eye and forehead. As he stumbles up the stairs (still shooting the handless pimp), the blood has changed, and finally in the shot where De Niro pantomimes shooting himself, the only blood on his face is on his chin. (01:39:40 - 01:42:40)
Answer: This is just my opinion. Remember Iris's line "Have you ever tried looking at your own eyeballs in the mirror?" or something to that effect. Well, Travis sees the madness reflected in his own eyes, doesn't like what he sees (as it reminds him of what he has done, what he might yet do).