Continuity mistake: In the hotel room, Easy Andy puts on the bed the two suitcases and opens them. Look at the very distinctive gun with the mother-of-pearl grip. It points left in opening shot and close-up, but when Andy picks the 44 Magnum up, the Model 36 is now pointed the opposite way. (00:54:45)
Continuity mistake: In the arms dealer scene, De Niro puts the last gun down with the others he is purchasing. It's at the exact angle as the others, pointing to the right, but in the close-up of the suitcase with the holster on top of the selection, now that gun (the so called "."380 Walther") is angled left, opposite of the others. (00:57:00)
Continuity mistake: Travis just fed to the security guy a phony name and address. He walks away. Look when the security guy urges the photographer to take a picture; behind the photographer there is a woman in a rather flashy orange dress, with orange headband and round sunglasses worn on top of that. But that same woman is also in front of Travis walking away, and one of the first to greet the senator. Likewise, when he shouts "Damn it!" a photographer with a brown shirt passes behind him, and it's the same one that passed behind Travis in the previous shot. (01:05:30)
Continuity mistake: Taking a fatal bullet from Travis, the robber falls back. He is not just breaking the snack display cabinet, but also knocking one of the dirty Spanish magazines off the clothesline. The magazine falls with him and gets all crimpled between his arm and the counter, but vanishes entirely in the rest of the scene. (01:08:50)
Continuity mistake: The robber is holding at gunpoint the store owner, arm outstretched. He turns around when Travis calls him, the arm still outstretched, straight, but when we see Travis blowing a hole in his face, that arm is bent, elbow close to the body. (01:08:50)
Continuity mistake: When the store owner beats the carcass of the robber, Travis' pistol is either to the right of the box on the counter or to the left, together with the gun of the robber. (01:09:30)
Continuity mistake: In the scene where De Niro shoots and kills the man robbing the convenience store, the clerk takes De Niro's gun off of him and starts beating the already dead body with a pole. If you watch the scene carefully, the dead body changes positions contrary to how the man is hitting him. Clearly the scene was shot several times with the dead body being placed in different positions. (01:09:40)
Continuity mistake: Once the store owner tells him he's gonna take care of everything and starts cursing in Spanish, Travis walks away from the store. You can see when Melio is bashing the Stick-Up Man that Travis left and nobody is in sight, but in the last overhead shot you can see someone's legs in the doorway. (01:09:40)
Continuity mistake: Listening to the music program after the robbery, Travis has the gun pointed up diagonally, leaning against the top of his head. Reverse shot and he has it pointed straight up, and only after a while it leans against his head. (01:10:45)
Continuity mistake: A policeman approaches Travis' taxi and tells him to leave; during that, as he is clapping his hands yelling "c'mon" in a super New Yorker way, a guy in a maroon shirt is pushing a cart behind the cop. Travis takes off in the next shot, and passes by, in front of him, the same guy that just went the other way. (01:12:50)
Continuity mistake: Travis plays with his foot on the TV and ends up breaking it. The TV set falls back in a dedicated view where it choreographically crashes at an angle and explodes. However, when we see De Niro sobbing over what happened, we also can get a peek at the very right of the frame and see a very much whole TV, lying parallel to the floor, screen still lit and working. (01:14:25)
Continuity mistake: Travis witnesses one guy beating another and dragging him away while he is in his car waiting for the arrival of the young prostitute. The corner is well lit by direct natural sunlight, and you can see the pavement, the working girls and the overweight woman walking along the building, all moving in the sun. Cut to De Niro approaching the girls, same street, roughly same spot including the graffiti landmark, and that sidewalk is in the shadow of the buildings. (01:15:10)
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)
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).