Gaffes

Joe's computer occasionally displays images of Liberty taken by a camera he has set up somewhere, presumably in his sniper's nest. But almost every shot from this camera is panning or tracking, and most are also obviously from a much lower angle than Joe's position. Some are actually looking up at Liberty from below.

After the SWAT team has cleared the park of pedestrians, there are several shots of Liberty standing by the cart with people walking by in the background.

Despite Liberty changing the direction she's facing several times during the initial stand-off with Joe, the red dot of his laser sight remains on her chest.

The multiple shots about sniper nest show, his rifle has set position to cover area around hot dog stand and covering area is very narrow because of opened only last bottom slot in louvers. However, when Liberty's husband killers (and later from CIA) start shooting on Liberty from building roof or top car park level, Joe found them in his sniper scope and hit them. The view from scope showing picture, that rifle is on same level or below of target, but for such angle sniper would must change angle of rifle to position, where are closed louvers. And he definitely would not move louvers, it would reveal his position. Especially when all SWAT snipers was around.

When Liberty first walks over to the hot dog stand and picks up the cuffs, the angle of the view through Joe's scope would imply that he's looking out of a 2nd or 3rd story window at most - which would be an utterly impractical angle for a sniper to be overlooking a crowded area.

Towards the end of the film we see Liberty drop her phone and headset on the ground but in the next seen we see the headset still in her ear.

Improperly worn rank insignia by the police "captain" (rank worn at odd angles). Perhaps LAPD has relaxed or arcane uniform standards, but the parallel captain's bars should either be worn "Army-style" (bars vertical) with the long edge of one bar centered along, parallel to, and 1/4" from the forward edge of the collar, or, "Navy-style" (bars diagonal) with the imaginary line between the two bars bisecting the angle of the collar's corner and lowest edge of the bars precisely 1 inch from the point of the collar.

Even though the red dot of Joe's laser-sight is on her chest, Liberty repeatedly scans buildings behind her while trying to spot Joe.