Let f : [ a , b ] → R be monotone increasing, F : [ a , b ] → R such that F ( x ) := ∫ [ a , x ] f and x 0 ∈ [ a , b ]. Then F is differentiable at x 0 if and only if f is continuous at x 0

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Let   f  :  [  a  ,  b  ]  →      R   be monotone increasing,   F  :  [  a  ,  b  ]  →      R   such that   F  (  x  )  :=      ∫          [      a      ,      x      ]        f and       x    0    ∈  [  a  ,  b  ]. Then   F is differentiable at       x    0   if and only if   f is continuous at       x    0

Rhett Pruitt · Accepted Answer

Suppose that   f is not continuous at       x    0  . Then, since   f is monotone increasing,       lim          x      →                                    x            0                          −              f  (  x  )  &amp;lt;  f  (      x    0    ) or       lim          x      →                                    x            0                          +              f  (  x  )  &amp;gt;  f  (      x    0    ). Let us suppose that the former assertion is correct and define   k  =      lim          x      →                                    x            0                          −              f  (  x  ). Then, if   x  &amp;lt;      x    0  ,            F      (      x      )      −      F      (              x        0            )              x      −              x        0              =                    ∫        x                              x            0                              f      (      t      )            d      t                      x        0            −      x        ⩽            k      (              x        0            −      x      )                      x        0            −      x        =  k  .On the other hand, if   x  &amp;gt;      x    0   then            F      (      x      )      −      F      (              x        0            )              x      −              x        0              =                    ∫                              x            0                          x            f      (      t      )            d      t              x      −              x        0              ⩾            f      (              x        0            )      (      x      −              x        0            )              x      −              x        0              =  f  (      x    0    )  .So, the limit of             F      (      x      )      −      F      (              x        0            )              x      −              x        0             at       x    0   does not exist and therefore   F is not differentiable at       x    0  .The case in which       lim          x      →                                    x            0                          +              f  (  x  )  &amp;gt;  f  (      x    0    ) is similar.

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