Continuity mistake: Near the start, when Slughorn shows Harry the framed photographs, the frames change position between the wideshot and close-ups.
Continuity mistake: As Harry and Luna come through Hogwart's gates, an Auror standing outside begins to walk backwards in through the gate. He makes it half-way through, but in the next shot, he is still at the end of the gate.
Continuity mistake: In the Room of Requirement, when Draco first uncovers the Vanishing Cabinet, the items on the floor behind him change position between the overhead shot and next close-up.
Continuity mistake: After drinking the poison in Slughorn's office, Ron falls and in the overhead shots the top of his head is near the rug's edge, but in side shots his head is well below the large diamond pattern, more than a foot away from the edge.
Continuity mistake: When Harry finds Hermione crying at the bottom of the stairs, with the conjured canaries flying overhead, there are numerous leaves/debris on the lowest steps and near her feet, but in the shots looking down from the top of the stairs the area is clear. (This has nothing to do with the few leaves that blow about.)
Visible crew/equipment: When Harry is lying outside the hut at the end of the film, Snape kicks his wand away. A block of neon tape can be seen briefly beneath the wand when Snape kicks it.
Audio problem: When Ron and Harry are walking down the corridor towards Potions class, Ron says, "...Quidditch trials coming up, I need to practice," but his mouth is not moving and/or not in sync with that dialogue.
Continuity mistake: In the Great Hall, when Ron sits down with Harry and Hermione before Quidditch, the newspaper Hermione is reading goes from her hand to under her arms, then back again, etc, between shots.
Continuity mistake: In the Great Hall before the Quidditch match, in the shots facing Hermione she is sitting opposite the empty place setting beside Ron, but in the shots facing Ron she is directly opposite him. Take note of the goblets, plate of eggs, etc.
Continuity mistake: In the scene before the Quidditch match, while Ron, Hermione and Ron are talking and Ron believes that Harry put something in his drink - in some angles Ginny is sitting directly next to Harry with their arms only an inch or two apart, in other angles Ginny's arm is nowhere to be seen, when they are sitting close enough that she should be visible.
Other mistake: In the film Harry Potter and the Half Blood Prince, Professor Slughorn (during one of the 'Slugclub' meetings) asks Hermione what her parents do for a living, she replies, 'They're Dentists.' Slughorn and the other wizards stare blankly, not knowing what Dentists are. However, in the film 'Harry Potter and the Chamber of Secrets' in the burrow, we see the magical clock that shows the family's whereabouts, and we see that there is a section marked 'Dentist'.
Other mistake: When Harry and his friends visit the Weasley's house the second time - when the Death Eaters attack - the constellation Scorpio is visible in the sky. Scorpio is a summer constellation. It shouldn't be visible in the winter. Furthermore, it probably can't be seen as far north as England (although I don't live there, so I'm not sure).
Continuity mistake: At the Three Broomsticks, when Slughorn and Harry speak, Slughorn holds a silver mug with a high foam head, but in the shot facing Harry, before Slughorn actually spills the drink the mug's foam head is gone (visible at the bottom of the screen), only to reappear when he spills it on Hermione ,whereupon the same previous angle is shown again without the foam.
Continuity mistake: During the first potion class scene during the movie, Hermione and Lavender are seen standing next to each other while being taught potions by Professor Slughorn. However, within the first minute or so of the lesson, Hermione and Lavender are standing far apart, then in the next shot, they are standing close together, with what looks like barely any space to move. They remain like this for the rest of the scene, but you can see the gap between the two within the first few minutes of the class.
Continuity mistake: The very first shot of Malfoy pulling the tapestry off the Vanishing Cabinet (after tossing a green apple in his hand) is used once again later (after Harry and Luna head off to Slughorn's party), when he is pulling the tapestry off again before placing a green apple in the Vanishing Cabinet. Note the same poof of dust, angle in which tapestry falls, and Draco's body position.
Continuity mistake: When Dumbledore and Harry enter the muggle house to find Slughorn the lights are all out; the only illumination comes from their wands. They enter the hallway, then both turn towards the living room with their wands in front of them. Despite this, the wall behind them is illuminated without either of them casting a shadow. The lighting in the hallway is obviously coming from a different source, not the wands. Can't be moonlight - it is a bright white light shining directly on the wall that was not visible when Harry and Dumbledore first entered the flat.
Continuity mistake: When Dumbledore and Harry approach the Slughorn armchair, on the left there is a white lampshade lying on the shelf hanging over the edge. When Slughorn stands up and walks to the door to lock it, the lampshade is farther back on the shelf.
Continuity mistake: When Ginny snatches 'Advanced Potion Making' from Harry in the common room, and says "Who's the Half-Blood Prince? . That's what it says right here: 'This book is the property of the Half-Blood Prince'", she opens it at a random page in the middle of the book. When Harry sees the words 'This book is the property of the Half-Blood Prince', they are written on the first page. Also, the continuity is wrong as she is shown to switch pages between very quick shots.
Continuity mistake: At the end when Harry moves the hair from Dumbledore's face in the closeup, several strands of hair are left. When they pan back to Dumbledore, different strands of hair are across his face.
Continuity mistake: When Ginny rips the Half-Blood Prince's potion book out of Harry's hand, before Harry gets up from leaning against the couch on the floor, Ginny is nowhere to be found in the chair on the very far left of the screen. As Harry walks around the couch while talking to Hermione, she is suddenly sitting in the same chair.
Answer: The purpose of the Pensieve is to preserve the memory exactly as it was recorded. Telling Harry something years after an event occurred is not as effective or as accurate as seeing. It is more impactful if Harry can watch events exactly as they unfolded. Regarding Slughorn, he had refused to give his complete memory because he was ashamed of the unintentional part he played in Voldemort's plan to create the Horcruxes. He was duped by Riddle, but he still did not want anyone to know how foolish he was.
raywest ★