Consider the following step function.

u$($x$)=$

$0$x $01$x $>0$

Calculate the average value of

u$($x$)$

on

$[1,1].$

Is there a value c in

$[1,1]$

such that

u$($c$)$

attains the average? If not, why does the mean value theorem for integrals fail?

No$,$ there is no such value of c because u$($x$)$ is not integrable on $[1,1].$Yes, u$($c$)$ attains the average value at c $=1.$Yes$,$ u$($c$)$ attains the average value at c $=0.$No$,$ there is no such value of c because u$($x$)$ is not continuous on $[1,1].$Yes, u$($c$)$ attains the average value at c $=0\mathrm{.}5.$