Continuous functions, believe it or not, are all sorts of useful. For one thing, they're the secret behind digital recording, including CDs and DVDs.
Here's a brief explanation of how continuous functions are used for recording. Suppose you want to use a digital recording device to record yourself singing in the shower. The song comes out as a continuous function. The digital recording device can't record what you sound like at every moment in time (there are infinitely many moments!), but it can record little bits of what you sound like several times a second (actually, way more often than that).
Since the song is a continuous function and continuous functions are nice (in all the ways we talked about earlier and in many other ways), the several-times-a-second recording contains enough information for a computer to reproduce more-or-less what you sounded like the whole time you were singing. If the little bits were recorded frequently enough and carefully enough, the reproduction will sound just like you.
Check out how Digital Recording works.
To see how mathematical this can be, check this one out.