Chapter 5, Problem 6RWDT

To determine

An example of an integrable function $f$ for which there is no value $c$ in $\left[a,b\right]$ such that the given condition is satisfied.

Answer to Problem 6RWDT

It has been determined that the function defined as $f\left(x\right)=-1$ for $x<0$ , $f\left(x\right)=0$ for $x=0$ and $f\left(x\right)=1$ for $x>0$ , that is, the sign function, is a required example.

Explanation of Solution

Given:

  $f\left(c\right)=\frac{1}{b-a}{\displaystyle {\int }_a^{b}f\left(x\right)dx}$

Concept used:

According to the rules of integration:

  ${\displaystyle {\int }_a^b{x}^{n}dx}={\left[\frac{x^{n+1}}{n+1}\right]}_a^b$

Calculation:

Let $a=-1$ and let $b=2$ .

Then, $\left[a,b\right]=\left[-1,2\right]$ .

Let $f\left(x\right)=-1$ for $x<0$ , $f\left(x\right)=0$ for $x=0$ and $f\left(x\right)=1$ for $x>0$ .

This is the sign function, which is discontinuous but integrable.

Put these values in $f\left(c\right)=\frac{1}{b-a}{\displaystyle {\int }_a^{b}f\left(x\right)dx}$ to get,

  $f\left(c\right)=\frac{1}{2-\left(-1\right)}{\displaystyle {\int }_{-1}^{2}f\left(x\right)dx}$

Simplifying,

  $f\left(c\right)=\frac{1}{3}{\displaystyle {\int }_{-1}^{2}f\left(x\right)dx}$

On further simplification,

  $f\left(c\right)=\frac{1}{3}\left({\displaystyle {\int }_{-1}^{0}f\left(x\right)dx}+{\displaystyle {\int }_0^{2}f\left(x\right)dx}\right)$

Continuing simplification,

  $f\left(c\right)=\frac{1}{3}\left(-{\displaystyle {\int }_{-1}^{0}dx}+{\displaystyle {\int }_0^{2}dx}\right)$

Solving,

  $f\left(c\right)=\frac{1}{3}\left(-\left[0-\left(-1\right)\right]+\left[2-0\right]\right)$

On further solving,

  $f\left(c\right)=\frac{1}{3}\left(-1+2\right)$

Finally,

  $f\left(c\right)=\frac{1}{3}$

Note that according to the function definition, there is no value $c$ in $\left[a,b\right]=\left[-1,2\right]$ such that $f\left(c\right)=\frac{1}{3}$ .

So, the assumed function $f$ is a required example.

Conclusion:

It has been determined that the function defined as $f\left(x\right)=-1$ for $x<0$ , $f\left(x\right)=0$ for $x=0$ and $f\left(x\right)=1$ for $x>0$ , that is, the sign function, is a required example.