Math Problems Vault

Problem 1.6.1

Suppose $\alpha$ increases on [ $a, b$ ], $a \leq x_{0} \leq b, \alpha$ is continuous at $x_{0}, f\left(x_{0}\right) = 1$ , and $f(x) = 0$ if $x \neq x_{0}$ . Prove that $f \in \mathscr {R}(\alpha )$ and that $\int f d \alpha = 0$ .

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Problem 1.6.2

Suppose $f \geq 0, f$ is continuous on [a,b], and $\int_{a}^{b} f(x) d x = 0$ . Prove that $f(x) = 0$ for all $x \in [a, b]$ . (Compare this with Exercise 1).

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Problem 1.6.3

(a)(b)(c)(d)

Define three functions $\beta_{1}, \beta_{2}, \beta_{3}$ as follows: $\beta_{j}(x) = 0$ if $x < 0, \beta_{j}(x) = 1$ if $x > 0$ for $j = 1,2,3$ ; and $\beta_{1}(0) = 0, \beta_{2}(0) = 1, \beta_{3}(0) = \frac{1}{2}$ . Let $f$ be a bounded function on $[-1,1]$ .

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(a)

Prove that $f \in \mathscr {R}\left(\beta_{1}\right)$ if and only if $f(0+) = f(0)$ and that then

\int f d \beta _{1} = f(0)

(b)

State and prove a similar result for $\beta_{2}$ .

(c)

Prove that $f \in \mathscr {R}\left(\beta_{3}\right)$ if and only if $f$ is continuous at 0.

(d)

If $f$ is continuous at 0 prove that

\int f d \beta _{1} = \int f d \beta _{2} = \int f d \beta _{3} = f(0)

Problem 1.6.4

If $f(x) = 0$ for all irrational $x, f(x) = 1$ for all rational $x$ , prove that $f \notin \mathscr {R}$ on $[a, b]$ for any $a < b$ .

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Problem 1.6.5

Suppose $f$ is a bounded real function on $[a, b]$ , and $f^{2} \in \mathscr {R}$ on $[a, b]$ . Does it follow that $f \in \mathscr {R}$ ? Does the answer change if we assume that $f^{3} \in \mathscr {R}$ ?

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Problem 1.6.6

Let $P$ be the Cantor set constructed in Sec. 2.44. Let $f$ be a bounded real function on $[0,1]$ which is continuous at every point outside $P$ . Prove that $f \in \mathscr {R}$ on $[0,1]$ . Hint: $P$ can be covered by finitely many segments whose total length can be made as small as desired. Proceed as in Theorem 6.10.

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Problem 1.6.7

(a)(b)

Suppose $f$ is a real function on $(0,1]$ and $f \in \mathscr {R}$ on $[c, 1]$ for every $c > 0$ . Define

\int _{0}^{1} f(x) d x = \lim _{c \rightarrow 0} \int _{c}^{1} f(x) d x

if this limit exists (and is finite).

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(a)

If $f \in \mathscr {R}$ on $[0,1]$ , show that this definition of the integral agrees with the old one.

(b)

Construct a function $f$ such that the above limit exists, although it fails to exist with $\left|f\right|$ in place of $f$ .

Problem 1.6.8

Suppose $f \in \mathscr {R}$ on $[a, b]$ for every $b > a$ where $a$ is fixed. Define

\int _{a}^{\infty } f(x) d x = \lim _{b \rightarrow \infty } \int _{a}^{b} f(x) d x

if this limit exists (and is finite). In that case, we say that the integral on the left converges. If it also converges after $f$ has been replaced by $\left|f\right|$ , it is said to converge absolutely.

Assume that $f(x) \geq 0$ and that $f$ decreases monotonically on $[1, \infty )$ . Prove that

\int _{1}^{\infty } f(x) d x

converges if and only if

\sum _{n = 1}^{\infty } f(n)

converges. (This is the so-called "integral test" for convergence of series).

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Problem 1.6.9

Show that integration by parts can sometimes be applied to the "improper" integrals defined in Exercises 7 and 8. (State appropriate hypotheses, formulate a theorem, and prove it). For instance show that

\int _{0}^{\infty } \frac{\cos x}{1+x} d x = \int _{0}^{\infty } \frac{\sin x}{(1+x)^{2}} d x

Show that one of these integrals converges absolutely, but that the other does not.

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Problem 1.6.10

(a)(b)(c)(d)

Let $p$ and $q$ be positive real numbers such that

\frac{1}{p}+\frac{1}{q} = 1

Prove the following statements.

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(a)

If $u \geq 0$ and $v \geq 0$ , then

u v \leq \frac{u^{p}}{p}+\frac{v^{q}}{q}

Equality holds if and only if $u^{p} = v^{q}$ .

(b)

If $f \in \mathscr {R}(\alpha ), g \in \mathscr {R}(\alpha ), f \geq 0, g \geq 0$ , and

\int _{a}^{b} f^{p} d \alpha = 1 = \int _{a}^{b} g^{q} d \alpha ,

then

\int _{a}^{b} f g d \alpha \leq 1

(c)

If $f$ and $g$ are complex functions in $\mathscr {R}(\alpha )$ , then

\left|\int _{a}^{b} f g d \alpha \right| \leq \left\{ \int _{a}^{b}\left|f\right|^{p} d \alpha \right\} ^{1 / p}\left\{ \int _{a}^{b}\left|g\right|^{q} d \alpha \right\} ^{1 / q}.

This is Hölder's inequality. When $p = q = 2$ it is usually called the Schwarz inequality. (Note that Theorem 1.35 is a very special case of this).

(d)

Show that Hölder's inequality is also true for the "improper" integrals described in Exercises 7 and 8.

Problem 1.6.11

Let $\alpha$ be a fixed increasing function on $[a, b]$ . For $u \in \mathscr {R}(\alpha )$ , define

\left\| u\right\| _{2} = \left\{ \int _{a}^{b}\left|u\right|^{2} d \alpha \right\} ^{1 / 2}

Suppose $f, g, h \in \mathscr {R}(\alpha )$ , and prove the triangle inequality

\left\| f-h\right\| _{2} \leq \left\| f-g\right\| _{2}+\left\| g-h\right\| _{2}

as a consequence of the Schwarz inequality, as in the proof of Theorem 1.37.

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Problem 1.6.12

With the notations of Exercise 11, suppose $f \in \mathscr {R}(\alpha )$ and $\varepsilon > 0$ . Prove that there exists a continuous function $g$ on $[a, b]$ such that $\left\| f-g\right\|_{2} < \varepsilon$ .

Hint: Let $P = \left\{ x_{0}, \ldots , x_{n}\right\}$ be a suitable partition of $[a, b]$ , define

g(t) = \frac{x_{i}-t}{\Delta x_{i}} f\left(x_{i-1}\right)+\frac{t-x_{i-1}}{\Delta x_{i}} f\left(x_{i}\right)

if $x_{i-1} \leq t \leq x_{i}$ .

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Problem 1.6.13

(a)(b)(c)(d)

Define

f(x) = \int _{x}^{x+1} \sin \left(t^{2}\right) d t

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(a)

Prove that $\left|f(x)\right| < 1 / x$ if $x > 0$ .

Hint: Put $t^{2} = u$ and integrate by parts, to show that $f(x)$ is equal to

\frac{\cos \left(x^{2}\right)}{2 x}-\frac{\cos \left[(x+1)^{2}\right]}{2(x+1)}-\int _{x^{2}}^{(x+1)^{2}} \frac{\cos u}{4 u^{3 / 2}} d u

Replace $\cos u$ by -1

(b)

Prove that

2 x f(x) = \cos \left(x^{2}\right)-\cos \left[(x+1)^{2}\right]+r(x)

where $\left|r(x)\right| < c / x$ and $c$ is a constant.

(c)

Find the upper and lower limits of $x f(x)$ , as $x \rightarrow \infty$ .

(d)

Does $\int_{0}^{\infty } \sin \left(t^{2}\right) d t$ converge?

Problem 1.6.14

Deal similarly with

f(x) = \int _{x}^{x+1} \sin \left(e^{t}\right) d t

Show that

e^{x}\left|f(x)\right| < 2

and that

e^{x} f(x) = \cos \left(e^{x}\right)-e^{-1} \cos \left(e^{x+1}\right)+r(x)

where $\left|r(x)\right| < C e^{-x}$ , for some constant $C$ .

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Problem 1.6.15

Suppose $f$ is a real, continuously differentiable function on [a, b], $f(a) = f(b) = 0$ , and

\int _{a}^{b} f^{2}(x) d x = 1

Prove that

\int _{a}^{b} x f(x) f^{\prime }(x) d x = -\frac{1}{2}

and that

\int _{a}^{b}\left[f^{\prime }(x)\right]^{2} d x \cdot \int _{a}^{b} x^{2} f^{2}(x) d x > \frac{1}{4}

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Problem 1.6.16

(a)(b)

For $1 < s < \infty$ , define

\zeta (s) = \sum _{n = 1}^{\infty } \frac{1}{n^{s}}

(This is Riemann's zeta function, of great importance in the study of the distribution of prime numbers). Prove that

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(a)

$\zeta (s) = s \int_{1}^{\infty } \frac{[x]}{x^{s+1}} d x$ and that

(b)

$\zeta (s) = \frac{s}{s-1}-s \int_{1}^{\infty } \frac{x-[x]}{x^{s+1}} d x$ , where $[x]$ denotes the greatest integer $\leq x$ . Prove that the integral in (b) converges for all $s > 0$ . Hint: To prove (a), compute the difference between the integral over [1,N] and the $N$ th partial sum of the series that defines $\zeta (s)$ .

Problem 1.6.17

Suppose $\alpha$ increases monotonically on $[a, b], g$ is continuous, and $g(x) = G^{\prime }(x)$ for $a \leq x \leq b$ . Prove that

\int _{a}^{b} \alpha (x) g(x) d x = G(b) \alpha (b)-G(a) \alpha (a)-\int _{a}^{b} G d \alpha .

Hint: Take $g$ real, without loss of generality. Given $P = \left\{ x_{0}, x_{1}, \ldots , x_{n}\right\}$ , choose $t_{i} \in \left(x_{i-1}, x_{i}\right)$ so that $g\left(t_{i}\right) \Delta x_{i} = G\left(x_{i}\right)-G\left(x_{i-1}\right)$ . Show that

\sum _{i = 1}^{n} \alpha \left(x_{i}\right) g\left(t_{i}\right) \Delta x_{i} = G(b) \alpha (b)-G(a) \alpha (a)-\sum _{i = 1}^{n} G\left(x_{i-1}\right) \Delta \alpha _{i}

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Problem 1.6.18

Let $\gamma_{1}, \gamma_{2}, \gamma_{3}$ be curves in the complex plane, defined on $[0,2 \pi ]$ by

\gamma _{1}(t) = e^{i t}, \quad \gamma _{2}(t) = e^{2 i t}, \quad \gamma _{3}(t) = e^{2 \pi i t \sin (1 / t)}

Show that these three curves have the same range, that $\gamma_{1}$ and $\gamma_{2}$ are rectifiable, that the length of $\gamma_{1}$ is $2 \pi$ , that the length of $\gamma_{2}$ is $4 \pi$ , and that $\gamma_{3}$ is not rectifiable.

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Problem 1.6.19

Let $\gamma_{1}$ be a curve in $\mathbb {R}^{k}$ , defined on $[a, b]$ ; let $\phi$ be a continuous 1-1 mapping of $[c, d]$ onto $[a, b]$ , such that $\phi (c) = a$ ; and define $\gamma_{2}(s) = \gamma_{1}(\phi (s))$ . Prove that $\gamma_{2}$ is an arc, a closed curve, or a rectifiable curve if and only if the same is true of $\gamma_{1}$ . Prove that $\gamma_{2}$ and $\gamma_{1}$ have the same length.

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