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Rogawski 6.2 #60 (Function that D/N Satisfy MVT for Integrals)

  1. Jan 23, 2008 #1
    1. The problem statement, all variables and given/known data
    Give an example of a function (necessarily discontinuous) that does not satisfy the conclusion of MVT for Integrals

    2. Relevant equations

    MVT for [tex]\int[/tex] = [tex]\frac{1}{b-a}[/tex][tex]\int ^{b}_{a}[/tex] f(x) dx

    3. The attempt at a solution

    So I should need one point of discontinuity on every interval that is a subset of the function's domain over [a,b] such that a<b, right?

    So couldn't...

    f(x) = {0 if x [tex]\leq[/tex] 1, 1 if x > 1} on the interval [0,2].

    We have: [tex]\frac{1}{2-0}[/tex][tex]\int^{2}_{0}[/tex] f(x) dx = 1/2, as f(x) does not equal 1/2.

    Would this work?
  2. jcsd
  3. Jan 23, 2008 #2


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    Homework Helper

    Your example looks correct.
  4. Jan 23, 2008 #3


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    You could do something as simple as changing the value at an endpoint:

    f(x)= x if [itex]0\le x< 1[/itex], f(1)= 1000.
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