When antidifferentiating, are we impliclty restricting to an interval?I was asked this question by a student I am tutoring and I was left a little puzzled because his textbook only defines antiderivatives on intervals (which leads me to believe its author would answer the question in the title in the affirmative).To my understanding, finding an antiderivative of f means finding a function F with F ′ = f. It does not matter if the domain of f is not connected. For example, ∫ d x x denotes an antiderivative on all of R − 0, and not just on some arbitrary interval I ⊆ R − 0.

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When antidifferentiating, are we impliclty restricting to an interval?I was asked this question by a student I am tutoring and I was left a little puzzled because his textbook only defines antiderivatives on intervals (which leads me to believe its author would answer the question in the title in the affirmative).To my understanding, finding an antiderivative of f means finding a function F with       F    ′    =  f. It does not matter if the domain of f is not connected. For example,   ∫            d      x        x   denotes an antiderivative on all of       R    −  0, and not just on some arbitrary interval   I  ⊆      R    −  0.

Elyse Huff · Accepted Answer

Step 1You can use whatever conventions you want; the author is free to choose hers and you are free to choose yours.One issue with defining antiderivatives on (nice) subsets of       R   which are not connected is that they are no longer just only unique up to constants; they are only unique up to locally constant functions. For example, on       R    ∖  {  0  } any function which has one constant value when x is negative and another when x is positive has zero derivative. This is the kind of subtlety that I suspect it would be a good idea to avoid in a calculus course.Step 2Another issue is that you&#039;d like to write antiderivatives down using definite integrals, but for example if a function f is not defined in the interval (−1,1) then it is unclear what an integral such as       ∫          −      2        x    f  (  x  )    d  x would mean for   x  &amp;gt;  1

When antidifferentiating, are we impliclty restricting to an interval? I was asked this question by

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