implicity

take the interval (0,1)
throw away the middle closed interval of length 1/2, leaving two open intervals
throw away the middle closed interval of length 1/3 that of the whole interval, from each of those two
leaving four shorter open intervals
throw away the middle closed interval of 1/4 of the length from each of those
then the middle 1/5 (closed; including endpoints) from the eight open intervals you have left
and so on
for 1/6, 1/7 ...

the points you have left are an uncountable, nowhere dense set of measure zero which isn't open, but is density open. which means each point of the set is a density point of the set. which is the purpose of the exercise.