Basically, yes. Ignore the public part; it just means they have a duty in general. A cop can watch you getting murdered in your house and they do not have to help you. The only exceptions for duty to an individual are the "danger creation exception" and the "special-relationship exception".
For 'danger creation exception', if the government doing or not doing something puts you in danger, they are liable. If you're already in danger it's not their fault.
For 'special relationship exception', basically, unless 'the government singles out a particular party [..] and affords that person special treatment', they have no liability to help that person or provide them proper services. The only exception to this is when the state has restricted the freedom of the individual, such as with prisoners.
This may be news to some of you, but that whole "I pay may taxes" line is ridiculous because of these conclusions. Yes you pay your taxes - to people who have no obligation to help you. You're welcome.
What's really odd about all this is the recent cases where someone who was handcuffed by police subsequently died. Technically this would fall under both these exceptions, because their freedom was restricted by the government and the police's action or inaction caused them danger. In the recent cases, though, the police were let off the hook. Weird.