Antiderivative of the greatest integer functionOne of my homework problem wants me to prove that the greatest integer function f ( x ) = ⌊ x ⌋ does not have an antiderivative. While thinking, I got to this expression, ∫ 0 x ⌊ t ⌋ d t = 1 2 ⌊ x ⌋ ( 2 x − ⌊ x ⌋ − 1 )Is there an antiderivative for the greatest integer function?The first part of the fundamental theorem of calculus implies the existence of an antiderivative for continuous functions. But what about discontinuous functions? Is there an antiderivative for any discontinuous functions?And thank you for any hints or ideas for my homework problem.

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Antiderivative of the greatest integer functionOne of my homework problem wants me to prove that the greatest integer function   f  (  x  )  =  ⌊  x  ⌋ does not have an antiderivative. While thinking, I got to this expression,       ∫    0    x    ⌊  t  ⌋    d  t  =      1    2    ⌊  x  ⌋  (  2  x  −  ⌊  x  ⌋  −  1  )Is there an antiderivative for the greatest integer function?The first part of the fundamental theorem of calculus implies the existence of an antiderivative for continuous functions. But what about discontinuous functions? Is there an antiderivative for any discontinuous functions?And thank you for any hints or ideas for my homework problem.

Braeden Shannon · Accepted Answer

Step 1The function   f  (  x  )  =  ⌊  x  ⌋ can be integrated, but it cannot have an antiderivative on the entire real line if we require that an antiderivative must be differentiable everywhere.Step 2Suppose, towards a contradiction, that F(x) were an antiderivative for f(x), differentiable everywhere. By Darboux&#039;s theorem, the derivative       F    ′    (  x  )  =  f  (  x  ) satisfies the intermediate value property, so since   f  (  0  )  =  0 and   f  (  1  )  =  1 there has to be some   a  ∈  (  0  ,  1  ) such that   f  (  a  )  =      1    2  . So   ⌊  a  ⌋  =      1    2  , but this is impossible.

Antiderivative of the greatest integer function One of my homework problem wants me to prove that t

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