Continuity mistake: When Salt and Winters first start talking between the glass window in the bunker Salt's body is facing slightly to the left. The camera cuts to the front of Salt and she is now facing right.
Continuity mistake: After Salt is arrested at the V.P.'s funeral, the police vehicles drive up a ramp directly behind the building with the Silvercup studios sign on the roof, yet when she is attempting to escape and rams the SUV she is in into the back of the escort patrol car in front of her, you can see they are now in front of the Silvercup studios building which is now a couple of blocks away in the background. (00:50:30 - 00:51:20)
Continuity mistake: In the scene where Salt is escaping the CIA for the first time, she removes her heels to be able to run, in the first shot her shoes are close together, but in the next shot they are further apart. (00:18:35)
Continuity mistake: The blood on Salt's face changes between shots, as the helicopter removes her from the White House lawn. (01:23:10)
Continuity mistake: Salt climbs down the elevator shaft dressed in a white shirt, but she exits with her skin and clothes totally spotless.
Continuity mistake: When Salt is arrested by the police, she is covered in a white powder. But, later, there is hardly any white powder to be seen.
Continuity mistake: Salt is in the Russian boat approaching the radioman. She throws some grenades down the stairs and some around, and explosions occur with fire and bits and pieces flying everywhere. Then she goes for a walk to kill some more guys and when she comes back to the same place it all looks nice and clean as before, no dust, no mess at all.
Continuity mistake: Salt leaves an ashtray with a pack of cigarettes on a table. Its position between the first angle and the following angle differs, with the pack suddenly standing out off the ashtray.
Continuity mistake: In the scene where Salt was escaping from her home and hid from her pursuer by dodging behind an air conditioner on the outside surface of the building, the previous scene shows that there's not enough space for hiding.
Continuity mistake: When Salt walks out of the church she is very dirty from the dust from the explosion. But in the scene in the police car afterwards she (and especially her hair) is completely clean.
Continuity mistake: When Salt escapes from the cop car, the broken windows and damage to the car appear/disappear during the chase, especially when her car jumps off the road and becomes a spotless new vehicle.
Continuity mistake: When interrogating Orlov her wedding ring is in different positions when the camera cuts to her.
Continuity mistake: When Salt is in the helicopter after leaving the White House, only the lock on the chain from the hands to the legs is unlocked, and yet when she runs out of the water, she has absolutely no arm or leg chains on her. (01:57:00)
Answer: They have to at least investigate that kind of an accusation, and when Salt so vigorously resists arrest it basically confirms, in their minds, that she is in fact a spy.
I actually kind of find it shocking that they would be willing to believe a guy who would do or say anything to save himself and since Salt was with the agency a long time, they could have at least given her the benefit of the doubt.