Chapter 5.4, Problem 71E

To determine

If the given statement is True or False.

Answer to Problem 71E

It has been determined that the given statement is TRUE.

Explanation of Solution

Given:

If $f$ is continuous on an open interval $I$ containing $a$ , then $F$ defined by $F\left(x\right)={\displaystyle {\int }_a^{x}f\left(t\right)dt}$ is continuous on $I$ .

Concept used:

According to The Fundamental Theorem of Calculus, if $f$ is continuous on $\left[a,b\right]$ and $F\left(x\right)={\displaystyle {\int }_a^{x}f\left(t\right)dt}$ , then $F^{\prime }\left(x\right)=f\left(x\right)$ for every point in $\left[a,b\right]$ .

If a function is differentiable on an interval, then it must be continuous on that interval.

Calculation:

Let $f$ be continuous on an open interval $I$ containing $a$ and let $F\left(x\right)={\displaystyle {\int }_a^{x}f\left(t\right)dt}$ .

Then, according to The Fundamental Theorem of Calculus, $F^{\prime }\left(x\right)=f\left(x\right)$ for every point in $I$ .

This implies that the function $F\left(x\right)$ is differentiable for every point in $I$ .

As discussed, the above implies that the function $F\left(x\right)$ is continuous at every point in $I$ .

So, if $f$ is continuous on an open interval $I$ containing $a$ , then $F$ defined by $F\left(x\right)={\displaystyle {\int }_a^{x}f\left(t\right)dt}$ is continuous on $I$ .

Hence, the given statement is TRUE.

Conclusion:

It has been determined that the given statement is TRUE.