We can only integrate real-valued functions that are reasonably well-behaved. No Dance Moms allowed. If we want to take the integral of f (x) on [a,b], there can't be any point in [a,b] where f zooms off to infinity. When it comes to definite integrals, this is bad:
Having a continuous function is great, but if it's only discontinuous at a few points, that's allowed too. For example, what if f (x) = 5 for all x ≠ 1 but is undefined at x = 1? That function looks like this:
If we want to integrate f from 0 to 2 there's one little spot where f isn't defined. That means the integral needs to account for all the area in this rectangle except for the line at x = 2:
Since a line doesn't have any area, taking out that line doesn't take away any area from the rectangle. This means it's not a problem for f to be undefined at that one point.
Similarly, it's not a problem for f to be undefined at ten separate points. Each individual line has no area, so the ten lines together have no area. It's also not a problem for f to be undefined at 100 separate points. Or a million.
When we can find the integral of a function on [a,b], we say that function is integrable on [a,b]. If a function is integrable for any interval we pick we say that function is integrable.