Is it true that if f is continuous, lim h → 0 1 h ∫ x x + h f ( t ) d t = f ( x )

Question

Is it true that if  f  is continuous,       lim          h      →      0            1    h        ∫          x              x      +      h        f  (  t  )  d  t  =  f  (  x  )

Alexia Hart · Accepted Answer

There is no need to invke the FTC for this. Note that, by the monotonicity of the integral,  h      min          x      ≤      s      ≤      x      +      h        f  (  s  )  =      ∫    x          x      +      h            min          x      ≤      s      ≤      x      +      h        f  (  s  )    d  t  ≤      ∫    x          x      +      h        f  (  t  )    d  tSimilarly,  h      max          x      ≤      s      ≤      x      +      h        f  (  s  )  =      ∫    x          x      +      h            max          x      ≤      s      ≤      x      +      h        f  (  s  )    d  t  ≥      ∫    x          x      +      h        f  (  t  )    d  tNow divide these inequalities by h and let   h  →  0. By continuity,      lim          h      →      0            min          x      ≤      s      ≤      x      +      h        f  (  s  )  =  f  (  x  )with a similar result for the max. Combinining the resulting inequalities yields the result.

Is it true that if f is continuous, <munder> <mo movablelimits="true" form="prefix">lim

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