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  1. #1
    Member Mollier's Avatar
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    Fundamental Theorem 1 & 2

    Hi, the way I understand it, the first part of the fundamental theorem of calculus says that every function that is continuous on the closed interval [a,b] has an antiderivative. If we denote the antiderivative by F it looks like,

    F(x) = \int^x_a f(t)dt.

    Now, F is continuous on [a,b] and differentiable on (a,b), and F'(x)=f(x) for all x in (a,b).
    Why is F not differentiable on the closed interval [a,b]?

    A corollary of this is that

    \int^b_a f(x)dx = F(b)-F(a).

    From what I read, the second part is almost the same as the first part but it does not assume continuity, but it assumes that an antiderivative of f exists.
    Does this mean that every continuous function has an antiderivative (part 1), but if a function has an antiderivative then it must not necessarily be continuous?

    I do not clearly see the difference between the corollary of part one, and theorem part 2.

    Thanks!
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  2. #2
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    Quote Originally Posted by Mollier View Post
    Hi, the way I understand it, the first part of the fundamental theorem of calculus says that every function that is continuous on the closed interval [a,b] has an antiderivative. If we denote the antiderivative by F it looks like,

    F(x) = \int^x_a f(t)dt.

    Now, F is continuous on [a,b] and differentiable on (a,b), and F'(x)=f(x) for all x in (a,b).
    Why is F not differentiable on the closed interval [a,b]?
    F is not (necessarily) differentiable at the end points, a and b, because that would require that the limits of the difference quotient from both directions be the same. And we have no information about F for x> b or for x< a.

    A corollary of this is that

    \int^b_a f(x)dx = F(b)-F(a).

    From what I read, the second part is almost the same as the first part but it does not assume continuity, but it assumes that an antiderivative of f exists.
    Does this mean that every continuous function has an antiderivative (part 1), but if a function has an antiderivative then it must not necessarily be continuous?
    Yes, if a function is continuous, then it has an anti-derivative. But that is NOT an "if and only if" theorem. There exist non-continuous functions that have anti-derivatives. In fact, take any "jump function", integrate it, and you get an example.
    If, say, f(x)= 0 for x< 0, f(x)= 1 for x\ge 0, then the antiderivative function is F(x)= 0 for x< 0, F(x)= x for x\ge 0.

    I do not clearly see the difference between the corollary of part one, and theorem part 2.

    Thanks!
    Last edited by CaptainBlack; October 8th 2010 at 09:23 PM. Reason: correct quote taggs
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