Given a set of daterange, finding the continuous range that includesa particular date

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So a client might leave a progrma and then return later, or they might simply switch to another staff_id.  (In which case one record will have and end date, and the next record will start on the next day.)  In this case I need to know "what period were they continuously in the program that includes X date?"  So I'd like to be able to do something like:

I've done this before with procedures specific to a particular table, and working with the start and end dates.  I'm now wanting to try to do this once generically that will work for all my cases.  So I'm hoping to do this in a way that performance isn't horrible.  And it's a little unclear to me how much and how I might be able to use the daterange operators to accomplish this efficiently.

Any advice or suggestions or ways to go about this appreciated.  Thanks!

Ken

p.s.,  Another small wrinkle is these records aren't always perfect, and ideally I'd allow for an optional fudge factor that would allow small gaps to be ignored.  I could just add that in every query (start_date+2,end_date-2), but it might be nice to have the function do it, if it didn't badly hurt performance.