
# 7.3: Integrability Conditions


## Proposition $$\PageIndex{1}$$

If $$a<b$$ and $$f:[a, b] \rightarrow \mathbb{R}$$ is monotonic, then $$f$$ is integrable on $$[a, b]$$.

Proof

Suppose $$f$$ is nondecreasing. Given $$\epsilon>0,$$ let $$n \in Z^{+}$$ be large enough that

$\frac{(f(b)-f(a))(b-a)}{n}<\epsilon .$

For $$i=0,1, \ldots, n,$$ let

$x_{i}=a+\frac{(b-a) i}{n}.$

Let $$P=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\} .$$ Then

\begin{aligned} U(f, P)-L(f, P) &=\sum_{i=1}^{n} f\left(x_{i}\right)\left(x_{i}-x_{i-1}\right)-\sum_{i=1}^{n} f\left(x_{i-1}\right)\left(x_{i}-x_{i-1}\right) \\ &=\sum_{i=1}^{n}\left(f\left(x_{i}\right)-f\left(x_{i-1}\right)\right) \frac{b-a}{n} \\ &=\frac{b-a}{n}\left(\left(f\left(x_{1}\right)-f\left(x_{0}\right)\right)+\left(f\left(x_{2}\right)-f\left(x_{1}\right)\right)+\cdots\right.\\ &\left.\quad+\left(f\left(x_{n-1}\right)-f\left(x_{n-2}\right)\right)+\left(f\left(x_{n}\right)-f\left(x_{n-1}\right)\right)\right) \\ &=\frac{b-a}{n}(f(b)-f(a)) \\ &<\epsilon . \end{aligned}

Hence $$f$$ is integrable on $$[a, b]$$. $$\quad$$ Q.E.D.

## Example $$\PageIndex{1}$$

Let $$\varphi: \mathbb{Q} \cap[0,1] \rightarrow \mathbb{Z}^{+}$$ be a one-to-one correspondence. Define $$f:[0,1] \rightarrow \mathbb{R}$$ by

$f(x)=\sum_{q \in \underset{q \leq x}{\mathbb{Q} \cap [0,1]}} \frac{1}{2^{\varphi(q)}.}$

Then $$f$$ is increasing on $$[0,1],$$ and hence integrable on $$[0,1]$$.

## Proposition $$\PageIndex{2}$$

If $$a<b$$ and $$f:[a, b] \rightarrow \mathbb{R}$$ is continuous, then $$f$$ is integrable on $$[a, b]$$.

Proof

Given $$\epsilon>0,$$ let

$\gamma=\frac{\epsilon}{b-a}.$

Since $$f$$ is uniformly continuous on $$[a, b],$$ we may choose $$\delta>0$$ such that

$|f(x)-f(y)|<\gamma$

whenever $$|x-y|<\delta .$$ Let $$P=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\}$$ be a partition with

$\sup \left\{\left|x_{i}-x_{i-1}\right|: i=1,2, \ldots, n\right\}<\delta .$

If, for $$i=1,2, \dots, n$$,

$m_{i}=\inf \left\{f(x): x_{i-1} \leq x \leq x_{i}\right\}$

and

$M_{i}=\sup \left\{f(x): x_{i-1} \leq x \leq x_{i}\right\},$

then $$M_{i}-m_{i}<\gamma .$$ Hence

\begin{aligned} U(f, P)-L(f, P) &=\sum_{i=1}^{n} M_{i}\left(x_{i}-x_{i-1}\right)-\sum_{i=1}^{n} m_{i}\left(x_{i}-x_{i-1}\right) \\ &=\sum_{i=1}^{n}\left(M_{i}-m_{i}\right)\left(x_{i}-x_{i-1}\right) \\ &<\gamma \sum_{i=1}^{n}\left(x_{i}-x_{i-1}\right) \\ &=\gamma(b-a) \\ &=\epsilon . \end{aligned}

Thus $$f$$ is integrable on $$[a, b]$$. $$\quad$$ Q.E.D.

## Exercise $$\PageIndex{1}$$

Suppose $$a<b, f:[a, b] \rightarrow \mathbb{R}$$ is bounded, and $$c \in[a, b] .$$ Show that if $$f$$ is continuous on $$[a, b] \backslash\{c\},$$ then $$f$$ is integrable on $$[a, b]$$.

## Exercise $$\PageIndex{2}$$

Suppose $$a<b$$ and $$f$$ is continuous on $$[a, b]$$ with $$f(x) \geq 0$$ for all $$x \in[a, b] .$$ Show that if

$\int_{a}^{b} f=0,$

then $$f(x)=0$$ for all $$x \in[a, b]$$.

## Exercise $$\PageIndex{3}$$

Suppose $$a<b$$ and $$f$$ is continuous on $$[a, b] .$$ For $$i=0,1, \ldots, n$$, $$n \in \mathbb{Z}^{+},$$ let

$x_{i}=a+\frac{(b-a) i}{n}$

and, for $$i=1,2, \ldots, n,$$ let $$c_{i} \in\left[x_{i-1}, x_{i}\right] .$$ Show that

$\int_{a}^{b} f=\lim _{n \rightarrow \infty} \frac{b-a}{n} \sum_{i=1}^{n} f\left(c_{i}\right).$

In the notation of Exercise $$7.3 .3,$$ we call the approximation

$\int_{a}^{b} f \approx \frac{b-a}{n} \sum_{i=1}^{n} f\left(c_{i}\right)$

a right-hand rule approximation if $$c_{i}=x_{i},$$ a left-hand rule approximation if $$c_{i}=x_{i-1},$$ and a midpoint rule approximation if

$c_{i}=\frac{x_{i-1}+x_{i}}{2}.$

These are basic ingredients in creating numerical approximations to integrals.