Chapter 5, Problem 7RWDT

To determine

An example of a function $f$ on $\left[a,b\right]$ , which satisfies the given condition.

Answer to Problem 7RWDT

It has been determined that a function $f$ , defined on the interval $\left[a,b\right]$ such that $f\left(x\right)=1$ if $x$ is rational and $f\left(x\right)=0$ if $x$ is irrational is a required example.

Explanation of Solution

Given:

The function $f$ is not the derivative of another function on $\left[a,b\right]$ .

Concept used:

According to the Fundamental Theorem of Calculus, Antiderivative part:

If the function $f$ is continuous on the interval $\left[a,b\right]$ , then the function $F\left(x\right)={\displaystyle {\int }_a^{x}f\left(t\right)dt}$ , which represents the accumulation of $f$ from $t=a$ to $t=x$ ; has a derivative at every point $x$ in $\left[a,b\right]$ , given by $F^{\prime }\left(x\right)=f\left(x\right)$ .

Calculation:

As discussed, every continuous function is a derivative of some accumulation function.

Let a function $f$ be defined on the interval $\left[a,b\right]$ such that $f\left(x\right)=1$ if $x$ is rational and $f\left(x\right)=0$ if $x$ is irrational.

Then, $f$ is a discontinuous function throughout the interval and more importantly, it is not the derivative of another function on $\left[a,b\right]$ .

Conclusion:

It has been determined that a function $f$ , defined on the interval $\left[a,b\right]$ such that $f\left(x\right)=1$ if $x$ is rational and $f\left(x\right)=0$ if $x$ is irrational is a required example.