In my case (fat PS3), yes it supports the "resume" function for Blu but not all movies are the same. It depends on individual movie if the "resume" is authored in the disc.
Example of movies that can resume:King Fu Hustle (SONY)
Fantastic Four & Ice Age 2 (Fox)
*UP (Disney)
These movies will resume to your last played position. You may also remove the disc from the PS3 player & insert it back at a later time & day.
*For the UP, you will have to access the main menu to resume the movie at the last stop.
Example of movies that can't resume:Transformers:RTF & GI Joe (Paramount)
Hulk, Hellboy 2, Jet Li's Fearless (Universal)
Open Season 2, Pirates Caribbean 2 (SONY)
Watching the Transformers: RTF is quite annoying & frustrating. Since the movie is almost 3 hours, hence I stopped it a few times (to attend work, house chores, etc) & unable to continue from my last played position. Each time I start the movie it will go back to the main menu.
It seems both Paramount & Universal don't support such function in their discs.