Are there any cases which the First Fundamental Theorem of Calculus would fail?

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aliasjuliankso9y · Accepted Answer

More simply take   f  (  x  )  =      {                            −          1          ,                                                                 0              ≤              x              &amp;lt;              1                              /                            2                                                            1          ,                                                                 1                              /                            2              ≤              x              ≤              1                                              . Then       F    ′    (  1      /    2  ) does not exist (write down thedefinition of the derivative as a limit of a difference quotient to see this).For your example, I don&#039;t quite follow your argument, but it seems you&#039;ve mentioned the necessary ingredients:   F is identically   0 (this essentially follows from the proof that your   f is integrable), so       F    ′   is identically   0. But       F    ′    (  x  )  ≠  f  (  x  ) for any rational number   x.

Are there any cases which the First Fundamental Theorem of

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