Looking through my old calc notes, I am reading that f(x) needs to be continuous on [a,b] and f ' (x) also needs to be continuous on [a,b]. Both conditions are needed. If you had to pick just one, then I'd say f(x) being continuous is much more important. Though I'm not 100% sure on this one. My thinking is that if there was any discontinuities on f(x), then the arc length would be distorted and overblown. The arc length should not account for any piece that isn't on the curve.
