A metaphor for discrete and continuous
And now for the extended example comparing continuous and discrete equations.
Here is a metaphor for continuous and discrete equations: Imagine you are driving down a country road in outback Queensland — the road is mostly flat, straight, and empty. You are doing all the things you learned in driver education, continually checking the road and your mirrors. You compensate immediately for any change in driving conditions. You are operating in a CONTINUOUS mode.
However, you start to get bored. So, seeing the road is nice and straight, you decide to catch up on some reading. Here’s the new strategy: glance at the road to make sure you’re headed in the right direction, look down at your book for 5 seconds. Glance up at the road again to readjust. Read for 5 more seconds, and so on. You’re in DISCRETE (or discontinuous) mode, and everything’s peachy. Your Δt, the amount of time between readjustments, is 5 seconds, as opposed to when you were continuously checking the road and your Δt was infinitely small (called dt).
But in a surprising geographic twist, suddenly the road becomes twisty, with cliffs on one side. And suddenly, your DISCRETE strategy of glancing at the road every 5 seconds doesn’t seem like a great idea. You might glance up just before a curve and make a sharp right, but you don’t see the following sharp left and go sailing off the cliff. Or you might glance up and see a straight patch ahead but fail to notice the sharp right ahead, so you go smashing into a wall of rock. In any case, you’d better decrease your Δt (look up more often) or better yet, go back to continuous mode!