Josh Zucker writes:

If f is continuous, isn't it automatically integrable?

Oops, you're right! I mis-stated my problem. Let's try that again, replacing the word "continuous" by the word "defined": Suppose that f is defined on the interval [a,b] and F is an antiderivative of f on (a,b) (that is, if F is a function that is continuous on [a,b] and differentiable on (a,b) with F'(x) = f(x) for all x in (a,b)). Can we conclude that f is integrable on [a,b]? Jim