Menu

Prove that if $f(x)$ is continuous on $[a,b]$ , then the function $F(x)=\int_a^x\,f(u)du$ is continuous on $[a,b]$ , differentiable on $(a,b)$ and its derivative is $F\,'(x)=f(x)$ for every $x$ in $(a,b)$ .

Primary tabs

Lesson Parent:

Integration III: Definite Integrals As Upper and Lower Sums

Problem:

Prove that if $f(x)$ is continuous on $[a,b]$ , then the function
$F(x)=\int_a^x\,f(u)du$
is continuous on $[a,b]$ , differentiable on $(a,b)$ and its derivative is $F\,'(x)=f(x)$ for every $x$ in $(a,b)$ .

Answer:

It is true that if $f(x)$ is continuous on $[a,b]$ , then the function $F(x)=\int_a^x\,f(u)du$ is continuous on $[a,b]$ , differentiable on $(a,b)$ and its derivative is $F\,'(x)=f(x)$ for every $x$ in $(a,b)$ .

Solution:

Before we start it is important to note that this proof will allow us to say that $F(x)$ is an antiderivative of $f(x)$ .

This proof is not overly complicated, but it is long. Here is an outline of the proof:

Pove that <Sign in to see all the formulas> by showing that:
- Recall that <Sign in to see all the formulas>.
- Restrict $x$ to the open interval $(a,b)$
- Prove the right-sided limit <Sign in to see all the formulas>.
- Prove the left-sided limit <Sign in to see all the formulas>.
- Because the right-sided limit and left-sided limits exist and are equal to each other we can say that <Sign in to see all the formulas> and therefore <Sign in to see all the formulas>.
Prove that F(x) is continuous:
- Because $F(x)$ is differentiable in $(a,b)$ we know that it is continuous in $(a,b)$ .
- Show that it is continuous from the right by proving <Sign in to see all the formulas>.
- Show that it is continuous from the left by proving <Sign in to see all the formulas>.
- Therefore, $F(x)$ is continuous on $[a,b]$ .

Let's first prove that <Sign in to see all the formulas>. Recall from our definition of a derivative that we need to prove that

<Sign in to see all the formulas>

To do this we our going to break the proof into proving the left-sided limit and the right-sided limit. We will prove that

<Sign in to see all the formulas>

and that

<Sign in to see all the formulas>

We know from this lesson that when the left-sided limit and right-sided limits are equal then we have proven the limit.

Since both the left and right-sided limits must exist we will restrict ourselves to $x$ in the open interval $(a,b)$ . To understand this more clearly think about what would happen if $x$ were equal to $a$ . We know nothing about $f(x)$ to the left of $a$ and therefore cannot determine if

<Sign in to see all the formulas>

is true. Similarly, if

$x=b$ we could not determine if

<Sign in to see all the formulas>

is true because we know nothing about

$f(x)$ to the right of

$b$ .

First, let's prove that

$M$ are the minimum and maximum values of

$f(x)$ in the closed interval <Sign in to see all the formulas>. Therefore,

$M$ are the minimum and maximum values of

$f(x)$ in the closed interval <Sign in to see all the formulas>. Now divide both sides by

$|h|$ to find that

$a$ from the right.

We know that

$b$ from the left.

This was a long proof, but we are finally done.

Prove that if f(x)f(x) is continuous on [a,b][a,b], then the function F(x)=∫xaf(u)du F(x)=\int_a^x\,f(u)du is continuous on [a,b][a,b], differentiable on (a,b)(a,b) and its derivative is F′(x)=f(x)F\,'(x)=f(x) for every xx in (a,b)(a,b).

Primary tabs

SAMPLE SOLUTION

GAIN AN ADVANTAGE

Prove that if $f(x)$ is continuous on $[a,b]$ , then the function $F(x)=\int_a^x\,f(u)du$ is continuous on $[a,b]$ , differentiable on $(a,b)$ and its derivative is $F\,'(x)=f(x)$ for every $x$ in $(a,b)$ .