How can I tell if my current is continuous in a first order DC circuit with a switch
... because that's literally the definition of an inductor!

$$v(t) = L \frac{\delta i(t)}{\delta t}$$

This means that for there to be a discontinuity in i(t), its derivative would have to be infinite, and so would the voltage across it. Since the voltage can only assume finite values, i(t) must be a continuous function. You can safely assume that the inductor current just after the switch opens is equal to the value just before it opens. That becomes the starting point for solving the behavior of the circuit from that point on.