If f is continuous on [a,b], then f is Darboux integrable hence Riemann integrable hence Lebesgue integrable on [a,b]. Hopefully my Advanced Calc II students will prove the first two facts within a few weeks. Scott Beaver James Propp wrote:
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
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