Mathematical Analysis
Leture 14
7.3 Basic properties
It is useful to define the integral $\ds\int_a^b f$ even if $a \not\lt b$. Suppose $b \lt a$ and $f \in \mathcal R[b,a]$, then define \begin{equation*} \int_a^b f \coloneqq - \int_b^a f . \end{equation*} For any function $f$, define $\ds \int_a^a f \coloneqq 0 . $
7.3 Basic properties
If the variable $x$ already has another meaning, then we may simply use a different letter. For example, \begin{equation*} \int_a^b f(s)\,ds \coloneqq \int_a^b f(x)\,dx . \end{equation*}
7.3 Basic properties
Theorem 7.3.1. (Linearity) Let $f$ and $g$ be in $\mathcal R[a,b]$ and $\alpha \in \R$.
- $\alpha \,f \in\mathcal R[a,b]$ and \begin{equation*} \int_a^b \alpha \,f(x) ~dx = \alpha \int_a^b f(x) ~dx . \end{equation*}
- $f+g \in\mathcal R[a,b]$ and \begin{equation*} \int_a^b \bigl( f(x)+g(x) \bigr) ~dx = \int_a^b f(x) ~dx + \int_a^b g(x) ~dx . \end{equation*}
Proof of Theorem 7.3.1, item 1
Consider $\alpha \geq 0$. Let $P$ be a partition of $[a,b]$, and $m_i \coloneqq \inf \bigl\{ f(x) : x \in [x_{i-1},x_i] \bigr\}$. As $\alpha \geq 0$,
$ \ds \inf \bigl\{ \alpha f(x) : x \in [x_{i-1},x_i] \bigr\}\qquad \qquad\qquad\qquad $
$= \alpha \inf \bigl\{ f(x) : x \in [x_{i-1},x_i] \bigr\} $ $ = \alpha m_i . $
Then $\, L(P,\alpha f) $ $\ds = \sum_{i=1}^n \alpha m_i \Delta x_i $ $ \ds = \alpha \sum_{i=1}^n m_i \Delta x_i $ $ \ds = \alpha L(P,f). $
Similarly $\, U(P,\alpha f) $ $ \ds = \alpha U(P,f). \qquad $
Proof of Theorem 7.3.1, item 1
Again, as $\alpha \geq 0$,
$ \ds \underline{\int_a^b} \alpha f(x)\,dx $ $\ds = \sup \, \bigl\{ L(P,\alpha f) : P \text{ a partition of } [a,b] \bigr\} $
$ \ds = \sup \, \bigl\{ \alpha L(P,f) : P \text{ a partition of } [a,b] \bigr\} $
$\ds = \alpha \, \sup \, \bigl\{ L(P,f) : P \text{ a partition of } [a,b] \bigr\} $
$\ds = \alpha \underline{\int_a^b} f(x)\,dx . \qquad\qquad\qquad\qquad\quad $
Similarly, we have $ \; \ds \overline{\int_a^b} \alpha f(x)\,dx = \alpha \overline{\int_a^b} f(x)\,dx . $
Proof of Theorem 7.3.1, item 1
Therefore, for $\alpha\geq 0$, $\alpha f\in \mathcal R[a,b]$ and \[ \int_a^b \alpha f(x)\,dx = \alpha \int_a^b f(x)\,dx . \]
To finish the proof of the first item (for $\alpha \lt 0$), we need to show that $-f$ is Riemann integrable and \[\int_a^b - f(x)\,dx = - \int_a^b f(x)\,dx.\] Try it! 📝 The proof of item 2 is also left as an Exercise 📝.
7.3 Basic properties
Theorem 7.3.2. Let $f \colon [a,b] \to \R$ and $g \colon [a,b] \to \R$ be bounded functions. Then
$\ds \overline{\int_a^b} (f+g) \leq \overline{\int_a^b}f+\overline{\int_a^b}g$
and $\ds\underline{\int_a^b} (f+g) \geq \underline{\int_a^b}f+\underline{\int_a^b}g.$
7.3 Basic properties
Theorem 7.3.3. (Monotonicity) Let $f \colon [a,b] \to \R$ and $g \colon [a,b] \to \R$ be bounded, and $f(x) \leq g(x)$ for all $x \in [a,b]$. Then
$\ds \underline{\int_a^b} f \leq \underline{\int_a^b} g\;\;$ and $\;\;\ds\overline{\int_a^b} f \leq \overline{\int_a^b} g . $
7.3.1 Continuous functions
Theorem 7.3.4. If $f \colon [a,b] \to \R$ is a continuous function, then $f \in \mathcal R[a,b]$.
7.3.1 Continuous functions
Theorem 7.3.4. If $f \colon [a,b] \to \R$ is a continuous function, then $f \in \mathcal R[a,b]$.
Proof. The first crucial observation is that because $f$ is continuous on a closed bounded interval, it is bounded and uniformly continuous. This means that given $\epsilon > 0$, there exists a $\delta > 0$ such that $$\abs{x-y} \lt \delta \Ra \abs{f(x)-f(y)} \lt \frac{\epsilon}{b-a}.$$ Now, let $P$ be a partition of $[a,b]$ where
$\Delta x_k = x_{k} - x_{k-1}\lt \delta\;$ for every subinterval of $P$.
7.3.1 Continuous functions
Theorem 7.3.4. If $f \colon [a,b] \to \R$ is a continuous function, then $f \in \mathcal R[a,b]$.
Proof. Given a particular subinterval $[x_{k-1},x_k]$ of $P$, we know from the Extreme Value Theorem (Thm 5.2.2) that the supremum $M_k = f(z_k)$ is attained for some $z_k \in [x_{k-1},x_k].$ And also the infimum $m_k$ is attained at some point $y_k \in [x_{k-1},x_k].$
7.3.1 Continuous functions
Theorem 7.3.4. If $f \colon [a,b] \to \R$ is a continuous function, then $f \in \mathcal R[a,b]$.
Proof. Given a particular subinterval $[x_{k-1},x_k]$ of P, we know from the Extreme Value Theorem (Thm 5.2.2) that the supremum $M_k = f(z_k)$ is attained for some $z_k \in [x_{k-1},x_k].$ And also the infimum $m_k$ is attained at some point $y_k \in [x_{k-1},x_k].$
7.3.1 Continuous functions
Theorem 7.3.4. If $f \colon [a,b] \to \R$ is a continuous function, then $f \in \mathcal R[a,b]$.
Proof. Given a particular subinterval $[x_{k-1},x_k]$ of P, we know from the Extreme Value Theorem (Thm 5.2.2) that the supremum $M_k = f(z_k)$ is attained for some $z_k \in [x_{k-1},x_k].$ And also the infimum $m_k$ is attained at some point $y_k \in [x_{k-1},x_k].$
$M_k-m_k $ $= f(z_k)-f(y_k)\quad$
$ \ds \lt \frac{\epsilon}{b-a} .$
7.3.1 Continuous functions
Theorem 7.3.4. If $f \colon [a,b] \to \R$ is a continuous function, then $f \in \mathcal R[a,b]$.
Proof. 👉 $\,M_k-m_k\lt \dfrac{\epsilon}{b-a} $. Consider now the difference
$\ds \overline{\int_a^b} f - \underline{\int_a^b} f $ $\leq U(P,f) - L(P,f) \qquad \qquad \qquad \qquad \quad$
$\ds = \left( \sum_{k=1}^n M_k \Delta x_k \right) - \left( \sum_{k=1}^n m_k \Delta x_k \right) $ $\ds = \sum_{k=1}^n (M_k-m_k) \Delta x_k$
$\ds \lt \frac{\epsilon}{b-a} \sum_{k=1}^n \Delta x_k$ $\ds = \frac{\epsilon}{b-a} (b-a) $ $\ds = \epsilon . \qquad \qquad \;\;\quad $
7.3.1 Continuous functions
Theorem 7.3.4. If $f \colon [a,b] \to \R$ is a continuous function, then $f \in \mathcal R[a,b]$.
Proof. Thus
$\ds \overline{\int_a^b} f - \underline{\int_a^b} f $ $\leq \epsilon .$
Since $\epsilon\gt 0 $ was arbitrary, \[ \overline{\int_a^b} f = \underline{\int_a^b} f \] and $f$ is Riemann integrable on $[a,b].$ $\;\bs$
7.3.1 Continuous functions
Bounded functions with finitely many discontinuities
Lemma 7.3.5. Let $f \colon [a,b] \to \R$ be a bounded function, $\{ a_n \}_{n=1}^\infty$ and $\{b_n \}_{n=1}^\infty$ be sequences such that $a \lt a_n \lt b_n \lt b$ for all $n$, with $\lim_{n\to\infty} a_n = a$ and $\lim_{n\to\infty} b_n = b$. Suppose $f \in \mathcal R[a_n,b_n]$ for all $n$. Then $f \in \mathcal R[a,b]$ and \begin{equation*} \int_a^b f = \lim_{n \to \infty} \int_{a_n}^{b_n} f . \end{equation*}
7.3.1 Continuous functions
Bounded functions with finitely many discontinuities
We say a function $f \colon [a,b] \to \R$ has finitely many discontinuities if there exists a finite set $$S = \{ x_1, x_2, \ldots, x_n \} \subset [a,b],$$ and $f$ is continuous at all points of $[a,b] \setminus S$.
Theorem 7.3.6. Let $f \colon [a,b] \to \R$ be a bounded function with finitely many discontinuities. Then $f \in \mathcal R[a,b]$.
7.3.1 Continuous functions
Bounded functions with finitely many discontinuities
Theorem 7.3.6. Let $f \colon [a,b] \to \R$ be a bounded function with finitely many discontinuities. Then $f \in \mathcal R[a,b]$.
7.3.1 Continuous functions
An example of a bounded function with a dense set of discontinuities
Thomae's function (Popcorn function): Riemann integrable.
More details: Wiki: Thomae's function or Thomae's book: Einleitung in die Theorie der bestimmten Integrale
7.3.1 Continuous functions
Another example of a bounded function with a dense set of discontinuities
Riemanns's example: $ \;\displaystyle f(x):=\sum_{n=1}^{\infty}\frac{(nx)}{n^2} $ where $(nx)= nx - \text{round} (nx) $.
7.3.2 More on integrable functions
Theorem 7.3.7. Let $f \colon [a,b] \to \R$ be Riemann integrable. Let $g \colon [a,b] \to \R$ be such that $f(x) = g(x)$ for all $x \in [a,b] \setminus S$, where $S$ is a finite set. Then $g$ is Riemann integrable and \begin{equation*} \int_a^b g = \int_a^b f. \end{equation*}
7.3.2 More on integrable functions
Theorem 7.3.8. Let $f:[a,b]\ra \R$ be a monotone function. Then $f\in \mathcal R[a,b]$.
The Integral: A Crux for Analysis
There are many theories of the integral.
The Riemann integral
is relatively simple, and it does the job nicely.
It is
certainly the most common and widely-used integral in
the world today.
Mathematicians, physicists, engineers,
and many others use the Riemann integral regularly.
The Integral: A Crux for Analysis
However, there exist examples of continuous function possessing a bounded derivative which is not Riemann integrable.
This example was given by the italian mathematician Vito Volterra in 1881.
The Integral: A Crux for Analysis
This example was surprising since it shows that within the context of Riemann's theory of integration the fundamental operations of differentiation and integration are not entirely reversible.
Thus, the process of differentiation could produce bounded functions $F'$ which fails to be integrable in Riemann's sense. In this case, the formula: \[ \int_{a}^{b}F'(x)~dx=F(b)-F(a) \] provided by the fundamental theorem of integral calculus, turns out to be meaningless.
The Integral: A Crux for Analysis
On the other hand, more advanced studies in analysis require an integral with better convergence properties. For example, if $f_n$ is a sequence of functions on a bounded interval $[a, b],$ then we would like to know that \[ \lim_{n\ra \infty} \int_a^b f_n = \int_a^b \lim_{n\ra \infty} f_n \qquad (*) \]
For the Riemann integral, the standard condition to guarantee (*) is uniform convergence of the sequence of functions $\{f_n \}$.
Garden of integrals
- The Riemann integral is based on subdividing the domain of the function (by way of a partition).
- The Lebesgue integral is based on subdividing the range of the function (you need measure theory).
- The Riemann-Stieltjes integral where the partition of a closed interval $I$ is determined with a monotonically increasing function defined on $I$.
- The Daniel integral which is equivalent to the Lebesgue integral but avoids the necessity of first developing measure theory.
- The Henstock-Kurzweil integral where a partition is measured in a more general way using a function called gauge.
- The Integral: A Crux for Analysis by Steven G. Krantz
- A Garden of Integrals by Frank E. Burk