Defining an Antiderivative for Monotone FunctionsI'm reading through a paper and I'm having trouble following the logic of the following step."Suppose ψ : ( 0 , ∞ ) → [ 0 , ∞ ) is a non-negative, non-decreasing function. Let Ψ be the primitive function of ψ, i.e. Ψ ′ = ψ.Can we actually do this? I believe monotonicity of ψ means that an antiderivative can indeed be defined, and would be continuous, but wouldn't it only be differentiable almost everywhere? In particular, at points of discontinuity of ψ (the set of which I believe will have measure zero, again by monotonicity) we cannot say Ψ ′ = ψ?In summary, is it instead true that Ψ ′ = ψ a.e.?

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Defining an Antiderivative for Monotone FunctionsI&#039;m reading through a paper and I&#039;m having trouble following the logic of the following step.&quot;Suppose   ψ  :  (  0  ,  ∞  )  →  [  0  ,  ∞  ) is a non-negative, non-decreasing function. Let   Ψ be the primitive function of   ψ, i.e.       Ψ    ′    =  ψ.Can we actually do this? I believe monotonicity of   ψ means that an antiderivative can indeed be defined, and would be continuous, but wouldn&#039;t it only be differentiable almost everywhere? In particular, at points of discontinuity of   ψ (the set of which I believe will have measure zero, again by monotonicity) we cannot say       Ψ    ′    =  ψ?In summary, is it instead true that       Ψ    ′    =  ψ a.e.?

Cristian Hamilton · Accepted Answer

Explanation:Yes you are right. The statement is in general false. Consider the monotonic function f on the reals such that   f  (  x  )  =  0 for   x  ≤  0 and   f  (  x  )  =  1  +  x for   x  &amp;gt;  0, if   f  =      F    ′   for some F for all x then f must have the intermediate value property, but it clearly doesn&#039;t.

Defining an Antiderivative for Monotone Functions I'm reading through a paper and I'm having troubl

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