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

[Hotsuma]

Homework Statement

Give an example of a function (necessarily discontinuous) that does not satisfy the conclusion of MVT for Integrals

Homework Equations

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

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?

 

[Dick]

Your example looks correct.

 

[HallsofIvy]

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.