When should you connect points on a coordinate graph?
This depends on the domain and/or range of the relation.
(1) If given a relation as a set of ordered pairs, you should not connect the points after you have plotted them. The implicit domain is just the input values expressed.
(2) If you are given a rule, but the implied range is restricted you might not connect the points. (Say the range is restricted to the natural numbers -- e.g. the number of people required to paint a room of a given size. Here the input would be an area, and the output would be the number of persons, which we expect to be a whole number)
(3) Some graphs have a jump discontinuity -- for example a step function like a charge rate for postal packages. Here you would not connect the endpoints, though many graphing utilities mistakenly do.
(4) If you are given a rule and the implied or explicit domain and range is the real numbers, you would connect the points. For instance, y=2x+3 gives a rule to find y given any input x, and unless stated otherwise x can be any real number.