[math-fun] another calculus question

Here's another question: Suppose that f is continuous 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]? (The version of the Fundamental Theorem of Calculus that Stewart gives states that, under the above hypothesis, plus the additional hypothesis that f is integrable on [a,b], we can conclude that the integral of f from a to b is F(b) - F(a). I'm asking whether the integrability of f really needs to be included as an extra assumption, or whether it follows from the other assumptions.) Thanks, Jim