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I don't know how do I have a well understanding for the below claim

Does it mean :

1/ The antiderivative dosn't exist

or

2/ The antiderivative exist and can't be expressed in elementary function (cosed form) or

3/The antiderivative should be exist and it is unkown

Claim:Every Continuous Function Have an Antiderivative

IV_
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zeraoulia rafik
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  • The second clarification of the claim is true: if you have a continuous function $f$, defined on a (possibly infinite) interval containing the point $a$, then $F(x)=\int_a^x f(y) dy$ is an antiderivative of $f$. It may or may not be possible to evaluate this integral in closed form, but it is guaranteed to exist (through the theory of either Riemann or Lebesgue integration) and to satisfy $F'=f$. – Ian Apr 03 '17 at 17:09
  • ok, could you give me some obstacle which forbid us to get the integral in closed form since it should be exist as you said – zeraoulia rafik Apr 03 '17 at 17:14
  • See http://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral – Ethan Bolker Apr 03 '17 at 17:20
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    How could the claim "Every continuous function has an antiderivative" possibly mean "The antiderivative doesn't exist"? Aren't these directly contradictory? – Alex Kruckman Apr 03 '17 at 17:20
  • @zeraouliarafik There is quite a bit of differential algebra in the way of such a development. At the same time this is a quite classic subject, so it should not be hard to find a detailed reference. A concrete thing to look up would be the Risch algorithm. – Ian Apr 03 '17 at 17:23
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    Did you learn about Galois theory? It can be used to proof that it's impossible to find a closed form solution for the degree 5 equation. There is also differential Galois theory which can be used to study which elementary functions have a closed form antiderivative. – mlainz Apr 03 '17 at 20:22

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To say that a ''Continuous Function Have an Antiderivative'' means that the antiderivative exists. maybe that we can express it in a ''closed form'' or not, but anyway it exists.

Emilio Novati
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