F is integrable but has no indefinite integralLet f ( x ) = { 0 , x ≠ 0 1 , x = 0. Then f is clearly integrable, yet has no antiderivative, on any interval containing 0, since any such antiderivative would have a constant value on each side of 0 and have slope 1 at 0 an impossibility.So does this mean that f has no indefinite integral?EDITMy understanding is that the indefinite integral of f is the family of all the antiderivatives of f, and conceptually requires some antiderivative to be defined on the entire domain. Is this correct?

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F is integrable but has no indefinite integralLet   f  (  x  )  =      {                            0          ,                                                                 x              ≠              0                                                            1          ,                                                    x              =              0.                                              Then f is clearly integrable, yet has no antiderivative, on any interval containing 0, since any such antiderivative would have a constant value on each side of 0 and have slope 1 at 0 an impossibility.So does this mean that f has no indefinite integral?EDITMy understanding is that the indefinite integral of f is the family of all the antiderivatives of f, and conceptually requires some antiderivative to be defined on the entire domain. Is this correct?

Sanaa Hinton · Accepted Answer

Step 1Attempting to express f&#039;s indefinite integral on its entire domain       R   runs into a technical problem at   x  =  0  ::  ∫  f  (  x  )        d    x  =      {                                        C            1                    ,                                                                 x              &amp;lt;              0                                                                        ?            ?            ?                    ,                                                    x              =              0                                                                        C            2                    ,                                                                 x              &amp;gt;              0                                                .Step 2No indefinite integral     ⟺   no antiderivative.On the other hand, f has an indefinite integral (and antiderivative) on every interval not containing 0.

Frank Day · Accepted Answer

Step 1This is a matter of definitions. Usually an antiderivative of a function f is any function whose derivative is f. The indefinite integral usually denotes the set of all antiderivatives.Since every derivative satisfies the intermediate values property, your function f cannot be a derivative and hence has no indefinite integral.Step 2A different thing is the integral function which might be defined as a function F such that      ∫    a    b    f  (  x  )  d  x  =  F  (  b  )  −  F  (  a  )  .This exists for all integrable functions.The fundamental theorem of Calculus states that the two concepts agree for continuous functions.

F is integrable but has no indefinite integral Let f ( x ) = { <mtabl

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