<Dl> <Dd> F (b) − F (a) = ∫ a b f (t) d t . (\ displaystyle F (b) - F (a) = \ int _ (a) ^ (b) f (t) \, dt .) </Dd> </Dl> <Dd> F (b) − F (a) = ∫ a b f (t) d t . (\ displaystyle F (b) - F (a) = \ int _ (a) ^ (b) f (t) \, dt .) </Dd> <P> This result may fail for continuous functions F that admit a derivative f (x) at almost every point x, as the example of the Cantor function shows . However, if F is absolutely continuous, it admits a derivative F ′ (x) at almost every point x, and moreover F ′ is integrable, with F (b) − F (a) equal to the integral of F ′ on (a, b). Conversely, if f is any integrable function, then F as given in the first formula will be absolutely continuous with F ′ = f a.e. </P> <P> The conditions of this theorem may again be relaxed by considering the integrals involved as Henstock--Kurzweil integrals . Specifically, if a continuous function F (x) admits a derivative f (x) at all but countably many points, then f (x) is Henstock--Kurzweil integrable and F (b) − F (a) is equal to the integral of f on (a, b). The difference here is that the integrability of f does not need to be assumed . (Bartle 2001, Thm . 4.7) </P>

When does the fundamental theorem of calculus fail
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