Continuity mistake: Near the beginning of the film, Llewyn goes into a subway station. The train that comes in is a different car model than the one he is sitting in in the next shot. Also, the exterior shot is of a train that did not yet exist in 1961. That model was introduced around 5-6 years later.
Inside Llewyn Davis (2013)
1 continuity mistake - chronological order
Directed by: Joel Coen, Ethan Coen
Starring: Oscar Isaac, Justin Timberlake, Carey Mulligan, Ethan Phillips
Factual error: In any scene where Llewyn is using a capo on his guitar, the capo he is using is a Shubb capo. They are excellent and he probably would have used one, save for the fact they were not invented until 1980, around eighteen years after the film takes place. It might be a Hamilton spring, but even they weren't around until later in the 60s.
Llewyn Davis: What is his thing?
Jim: Huh?
Llewyn Davis: Him. Troy.
Jim: Wonderful performer.
Llewyn Davis: Is he?
Jim: Wonderful.
Llewyn Davis: Does he have a higher function?
Trivia: Near the start, Llewyn looks at the back of his album made with Mike Timlin. The biographical information below Mike, however, talks about Dave Van Ronk, the folk singer about whom the movie is loosely based. (00:06:48)