Continuity mistake: Presumably the same day when he passes by Palantine's HQ and does not see Betsy at her desk, Travis nearly bumps his taxi into the same girl (Jodie Foster) he failed to save from her pimp. The sequence does not concern itself with continuity at all; the girls move away from the taxi twice in two different ways (look at how the long haired hooker turns), and De Niro's hands change position on the wheel with no logic. (00:51:40)
Continuity mistake: When Travis joins Easy Andy in the cab, cars and people in the street behind him change in the reverse angle. (00:54:00)
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)
Revealing mistake: When Travis starts spelling his phony name out to the security guy, a ginger kid walks in, stares at the camera, and hides away as quick as he came. In similar fashion, another kid with a red baseball cap looks at the camera behind De Niro once this walks by evading the photographer. (01:04:20)
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)
Other mistake: Travis Bickle's voiceover reading the letter to Mom and Dad says that July is Father's day month. But the holiday happens in the month of June (at least in the US). On the other hand, it's more than just a Character Error, because when you see the written part of the card held by Robert De Niro, the word used is "June" and not "July." (01:12:00 - 01:13:20)
Revealing mistake: During De Niro's voiceover in the "Dear father and mother" letter, the camera pans towards the stage where Senator Charles Palantine is giving his speech. As De Niro mentions the anniversary, a dude in the crowd spots the camera and waves hi to it, says something like "Wonder what's up?" to someone next to him in the crowd and leans to his left trying to stay in frame longer. (01:12:05)
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)
Visible crew/equipment: Travis clears out because the policeman told him he can't park there and so he starts the car and drives. Everyone in the crowd looks (many turn around explicitly and ) as he passes by, in particular looking right where the camera car in front of the taxi is. The senator is supposed to talk in a different direction, where De Niro is looking. (01:12:55)
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)
Other mistake: When Travis is negotiating with Sport for Iris' services, an off-screen crew member can be heard saying, "but no rough stuff" just before Sport says it. (01:17:00)
Revealing mistake: After taking Iris out for breakfast, Travis is in his car and looks at the building when Iris lives and/or works. And in that establishing shot, that lasts about a second fading almost immediately, from his car he is watching...himself, because it is re-used footage from the scene of one day earlier, when he was entering the building to pay for her service. The person you see go through the door, is De Niro, same shirt and jeans of that scene. Not only that, most people at the window are the same and in the exact same position, and the passersby in the street are the same. (01:18:30 - 01:29:30)
Answer: It helps him create a tolerance for pain. As we see, he certainly gets a lot of that dished out to him.