Continuity mistake: The only letter that Alexander B. Cust really wrote himself is briefly shown when Poirot visits the office of the Twinkle Toes stocking company. The lowercase "R" is too high every time (meaning that the typewriter isn't perfect, a detail put in for realism). Yet all the close-ups of a typewriter progressing through the work on that letter, or any of the letters written on the same machine, show a perfectly aligned "r."
Continuity mistake: When the first Guadalajara brother is shot, and his body is on the floor, you can see the blood on the glass. Two seconds later, in a close-up shot, it is missing, only to reappear a few seconds later.
Episode #1.1 - S1-E1
Continuity mistake: When David meets the Home Secretary for the first time outside her car, he is wearing a grey shirt. In the next scene (in the lobby) his shirt is white, before turning grey again later.
Continuity mistake: Ruth Deaver is helping Alan Pangborn put on his tie. The camera is facing Ruth's back and we see Alan's face in the mirror. The camera then switches angles to face Alan's back and Ruth's face. Alan's collar which was popped straight up from the first angle is now perfectly down over his newly tied tie from the second angle. (00:24:55)
Continuity mistake: John Paul Getty approaches the kidnapper's cars and Getty removes his sunglasses and tucks them into his coat pocket. The next shot shows him still wearing them as the kidnappers approach.
Continuity mistake: The man who has fallen down the ravine raises his hand as he shouts for the EMTs. The scene immediately cuts to a shot of him holding his wound with his hand down.