Differentiation and Integration

Calculate all Dini derivatives of $\chi_{\mathbb {Q}}$, the characteristic function of rationals.

Let

Calculate all four Dini derivatives of $f$ at $x=0$ and at $x=1$.

Calculate all Dini derivatives of $x \chi_{\mathbb {Q}}(x)-x \chi_{\mathbb {Q}^{C}}(x)$ at $x=0$ and at $x=1$.

Let $f(x)=2^{-n}$ if $x \in \left(2^{-n}, 2^{-n+1}\right]$ and $f(0)=0$. Calculate all four Dini derivatives of $f$ at $x=0$.

Calculate all Dini derivatives of $\chi_{[0,1]}(x)$ at $x=0$ and at $x=1$.

Construct an increasing function $f:[-1,1] \rightarrow \mathbb {R}$ such that none of the four Dini derivatives at 0 are equal.

Prove that a function $f$ is differentiable at $x_{0}$ (in the standard meaning) if and only if all 4 Dini derivatives are equal and finite, in which case $f^{\prime }\left(x_{0}\right)= D^{+} f\left(x_{0}\right)=D_{+} f\left(x_{0}\right)=D^{-} f\left(x_{0}\right)=D_{-} f\left(x_{0}\right)$.

If $f$ is defined and finite on $[-c, c]$, prove the following:

If $f$ and $g$ are defined and finite on $[a, b]$, prove that for every $x, D^{+}(f+ g)(x) \leq D^{+} f(x)+D^{+} g(x)$. Give an example to show that the inequality may be strict.

Let $f, g: \mathbb {R} \rightarrow \mathbb {R}$ be continuous and suppose that for every $x \in \mathbb {R}$, $D^{+} f(x)=D^{+} g(x)<\infty$. Prove that $f$ and $g$ differ by a constant.

Give an example to show that the chain rule for Dini derivatives $D^{+}(f \circ g)(x)=D^{+} f(g(x)) D^{+} g(x)$ does not hold.

Let $A$ be a subset of $\mathbb {R}$ of measure 0 . Prove that there exists a continuous increasing function $f: \mathbb {R} \rightarrow \mathbb {R}$ for which each Dini derivative is infinite on each point of $A$.

Let $f:(a, b) \rightarrow \mathbb {R}$ be a continuous function. Prove that $f$ is nonincreasing on ( $a, b$ ) if and only if $D^{+} f \leq 0$ on ( $a, b$ ).

Let $f, g:[a, b] \rightarrow \mathbb {R}$ be continuous functions such that $D^{+} f(t) \leq 0$ and $D^{+} g(t) \leq 0$ on $(a, b)$. If $h(t)=\max \left\{ f(t), g(t)\right\}$, prove that $D^{+} h(t) \leq 0$ on $(a, b)$.

If $f$ and $g$ are defined and finite on $[a, b]$, prove the following:

Let $f$ and $g$ be nonnegative and continuous at $c$.

Complete the proof of Proposition 9.1.4 (if $f$ is monotone, the four Dini derivatives of $f$ are measurable) by showing that if $f$ is monotone then the remaining three Dini derivatives of $f$ are measurable ( $D_{+}, D^{-}$, and $D_{-}$).

Suppose that $D_{+} f\left(x_{0}\right)$ is finite. Prove that $f$ is continuous from the right at $x_{0}$.

Suppose $f$ is continuous on $(a, b)$. Prove that $f$ is increasing on $(a, b)$ if and only if at least one of the four Dini derivatives of $f$ is nonnegative on $(a, b)$.

Let $f$ be an increasing function on $[a, b]$ and let $x_{0} \in (a, b)$. Prove that none of the Dini derivatives can be negative (or equal to $-\infty$ ) at $x_{0}$.

(Dini, 1878) Let $f:[a, b] \rightarrow \mathbb {R}$ be a continuous function. Prove that the bounds of each of the Dini derivatives are equal to the bounds of the difference quotient

as $x, y \in [a, b]$. Use this to conclude that if a single Dini derivative of a continuous function $f$ is continuous at a point $x_{0}$, then so are the other three. Show that in this case all four Dini derivatives are equal, so $f$ is differentiable at $x_{0}$.

Give an example to show that the conclusions of Exercise 9.1.21 need not hold if the continuity assumption on $f$ is dropped.

Let $f$ be continuous on $[a, b]$, and let a constants $M$ and a set $E \subset [a, b]$ of positive measure satisfy $D^{+} f(x) \geq M$ for all $x \in E$ (or $D^{-} f(x) \geq M$ for all $x \in E$ ). Prove that for every $\varepsilon >0$ there exists a finite collection of disjoint intervals $\left(a_{1}, b_{1}\right), \ldots ,\left(a_{n}, b_{n}\right)$ in $[a, b]$ such that $\sum_{i=1}^{n}\left(b_{i}-a_{i}\right)> m(E)-\varepsilon$ and

Let $f$ be a real-valued function. Prove that the set of points where $D^{+} f(x)<D_{-} f(x)$ is countable.

Give an example showing that it is possible that the set $A_{1}$ in Theorem 9.1.7 have Lebesgue measure $b-a$. Do the same for sets $A_{2}$ and $A_{3}$.

Theorem 9.1.7 (The Denjoy--Young--Saks Theorem): Let $f$ be a real-valued function defined on $[a, b]$. Then almost every point $x \in [a,b]$ is in one of four sets: $A_1$, on which $f$ has a finite derivative; $A_2$, on which $D^{+} f=D_{-} f$ is finite, $D^{-} f=\infty$, and $D_{+} f=-\infty$; $A_3$, on which $D^{-} f=D_{+} f$ is finite, $D^{+} f=\infty$, and $D_{-} f=-\infty$; $A_4$, on which $D^{-} f=D^{+} f=\infty$ and $D_{-} f=D_{+} f=-\infty$.

Let $f$ be continuous on $[a, b]$ and suppose that it has a local maximum at $c \in (a, b)$. Prove that

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be continuous and suppose that $D^{+} f$ is Riemann integrable over $[a, b]$. Prove that

If $f$ is continuous on $[a, b]$ prove that the four Dini derivatives of $f$ are measurable.

If $f$ is measurable prove that the four Dini derivatives of $f$ are measurable.

A function $f \in C[a, b]$ is called Hölder continuous of order $\delta$, notation $f \in \Lambda_{\delta }$, if there is a constant $C$ such that $\left|f(x)-f(y)\right| \leq C\left|x-y\right|^{\delta }$ for all $x, y \in [a, b]$. Prove that the Hölder continuous functions form a set of first categories in $C[a, b]$.

Let $\mathcal{F}=\left\{ [p, q]: 0 \leq p<q \leq 1, p, q \in \mathbb {Q}\right\}$. Prove that $\mathcal{F}$ is a Vitali cover for $[0,1]$.

Let $E \subset \mathbb {R}$, and for every $x \in E$ let $\left\{ \beta_{n}(x)\right\}$ be a strictly decreasing sequence of real numbers that converges to $x$. Prove that $\mathcal{F}=\left\{ \left[x, \beta_{n}(x)\right]: x \in E\right\}$ is a Vitali cover for $E$.

Let $E \subset \mathbb {R}$ be a union (not necessarily countable and not necessarily disjoint) of nondegenerate intervals (open, closed, half open and half closed, infinite, etc). Show that $E$ is measurable.

Show that the Vitali Covering Lemma extends to the case in which the covering collection consists of nondegenerate intervals that are not necessarily closed.

Show that the Vitali Covering Lemma does not extend to the case in which the covering collection contains some intervals that are not necessarily nondegenerate.

Prove that a set $E \subset \mathbb {R}$ has measure 0 if and only if there is a countable collection of open intervals $\left\{ I_{k}\right\}_{k=1}^{\infty }$ for which each point in $E$ belongs to infinitely many of these intervals and $\sum_{k=1}^{\infty } \ell \left(I_{k}\right)<\infty$.

Prove that each function $g_{n}$ defined in the proof of Corollary 9.2.9 is Riemann integrable.

In the proof of Corollary 9.2.9 (which states that for $f$ increasing on $[a,b]$, $f'$ is integrable over $[a,b]$ and $\int_{a}^{b} f' \leq f(b) - f(a)$), $f$ is first extended to $[a, b+1]$ by $f(x) = f(b)$ for $x \in [b, b+1]$, and then $g_{n}(x) := \dfrac {f(x+1/n) - f(x)}{1/n}$ for $x \in (a, b)$.

Prove inequality (9.23).

Inequality (9.23): Let $f$ be an increasing function on $[a, b]$, extended to the interval $[b, b+1]$ by $f(x)=f(b)$ for $x \in [b, b+1]$, and let

Then

Prove Theorem 9.2.12.

Theorem 9.2.12: Let $f$ be a monotone function on $(a,b)$. Then $f$ has at most a countable number of jump discontinuities on $(a,b)$.

Prove Theorem 9.2.13.

Theorem 9.2.13: Let $\left\{ x_{n}\right\}_{n=1}^{\infty }$ be a countable subset of $[a,b]$ and let $\left\{ c_{n}\right\}_{n=1}^{\infty }$ be a sequence of positive numbers for which the series $\sum_{n=1}^{\infty } c_{n}$ converges. Given $x \in [a,b]$, let $E(x)$ be the subset of $\mathbb {N}$ consisting of those integers $n$ for which $x_{n} < x$, and let $f(x) := \sum_{n \in E(x)} c_{n}$ (with $f(x) = 0$ if $E(x) = \varnothing$). Then: (a) $f$ is a monotone-increasing function on $[a,b]$. (b) $f$ has a jump discontinuity at each $x_{n}$. (c) The jump at $x = x_{n}$ is $c_{n}$. (d) $f$ is continuous at every other point of $[a,b]$.

Let $E$ be a subset of $\mathbb {R}$ and $A$ a countable subset of $E$. Prove or disprove: There is a monotone function on $E$ that is continuous only at points in $E \backslash A$.

Let $f$ be an increasing function on $[a, b]$. Prove that

Prove inequality (9.20).

Inequality (9.20): If $f$ is an increasing function on $(a, b)$, then

This is the second half of Lemma 9.2.7, whose first half reads: Let $f$ be an increasing function on $[a, b]$ and $T=\left\{ x \in (a, b): D^{+} f(x)>D_{-} f(x)\right\}$. Then $m^{*}(T)=0$.

Definition 9.1.1: For a real-valued function $f$ on $[a, b]$ and $x_{0} \in (a, b)$, the Dini derivatives are $D^{+} f\left(x_{0}\right)=\limsup_{x \rightarrow x_{0}+} \frac{f(x)-f\left(x_{0}\right)}{x-x_{0}}$, $D_{+} f\left(x_{0}\right)=\liminf_{x \rightarrow x_{0}+} \frac{f(x)-f\left(x_{0}\right)}{x-x_{0}}$, $D^{-} f\left(x_{0}\right)=\limsup_{x \rightarrow x_{0}-} \frac{f(x)-f\left(x_{0}\right)}{x-x_{0}}$, $D_{-} f\left(x_{0}\right)=\liminf_{x \rightarrow x_{0}-} \frac{f(x)-f\left(x_{0}\right)}{x-x_{0}}$.

Let $E \subset [a, b]$ be a set of measure 0 . Prove that there exists a continuous increasing function $f$ such that $f^{\prime }(x)=+\infty$ for all $x \in E$.

Prove or disprove: the derivative of a continuous, monotone function is differentiable a.e.

Let $f$ be continuously on $(a, b)$ and let $E=\left\{ x \in (a, b): D_{-} f(x)< D^{+} f(x)\right\}$. Prove that $E$ cannot contain a nondegenerate interval.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be monotone increasing and let $g(x)=\lim_{h \rightarrow 0+} f(x+ h)$. Prove that the function $g$ has the following properties:

Let $f:[a, b] \rightarrow \mathbb {R}$ be strictly increasing and $E \subset [a, b]$ such that at every point $x \in E, D^{+} f(x) \leq M$. Prove that $m^{*} f(E) \leq M m^{*}(E)$.

Let $f:[a, b] \rightarrow \mathbb {R}$ be strictly increasing and $E \subset [a, b]$ such that at every point $x \in E, D_{-} f(x) \geq C$. Prove that $m^{*} f(E) \geq C m^{*}(E)$.

Prove that every monotone function defined on an interval is Borel measurable.

Let $f$ be continuous on $\mathbb {R}$. Prove or disprove: there exists an open interval on which $f$ is monotone.

Let $f$ be differentiable on $\mathbb {R}$, let $x_{0} \in \mathbb {R}$, and suppose that $f^{\prime }\left(x_{0}\right)>0$. Prove or disprove: there exists an open interval $I$ containing $x_{0}$ such that $f$ is monotone on $I$.

(Fubini) Let $f_{k}:[a, b] \rightarrow \mathbb {R}$ be monotone increasing functions for $k= 1,2, \ldots$ and assume that the series $f(x)=\sum_{k=1}^{\infty } f_{k}(x)$ converges pointwise on $[a, b]$. Prove that $f^{\prime }(x)=\sum_{k=1}^{\infty } f_{k}^{\prime }(x)$ for almost all $x \in [a, b]$.

Let $f_{k}:(0,1) \rightarrow \mathbb {R}$ be monotone increasing for $k=1,2, \ldots$ and suppose that $\lim_{n} f_{n}(x)=1$ for a.e. $x \in (0,1)$. Prove that $\liminf_{n} f_{n}^{\prime }(x)=0$ for a.e. $x \in (0,1)$.

Give an example of functions $f, g:[0,1] \rightarrow \mathbb {R}$ such that $f$ is increasing and continuous, $g$ is a polynomial, and

Prove or disprove: if $f: \mathbb {R}^{2} \rightarrow \mathbb {R}$ is increasing in each variable then it is Borel measurable.

Let $\phi$ denote the Cantor-Lebesgue function and set $\phi (x)=0$ for $x<0$ and $\phi (x)=1$ for $x>1$. Let $\left\{ \left[a_{n}, b_{n}\right]\right\}$ be the collection of all closed subintervals of $[0,1]$ with rational endpoints, let

and let $G=\sum_{1}^{\infty } 2^{-n} F_{n}$. Prove that $G$ is continuous and strictly increasing on $[0,1]$ and that $G^{\prime }=0$ a.e.

The purpose of this exercise is to establish the so-called Helly Selection Principle: If $\left\{ f_{n}\right\}$ is a uniformly bounded sequence of increasing functions on $[0,1]$ then there exists a subsequence $\left\{ f_{n_{k}}\right\}$ that converges pointwise at every point of $[0,1]$ to an increasing function $f$.

Let $\left\{ I_{n}\right\}$ be a sequence of subintervals of $[0,1]$ with rational endpoints. For each $I_{n}$, let $E_{n}$ be the set of all continuous functions on $[0,1]$ that are monotone on $I_{n}$. Using these sets proves that there exists a continuous function on $[0,1]$ that is not monotone on any subinterval of $[0,1]$.

Let $A \subset [0,1]$ be a set of the first category, and $f: A \rightarrow \mathbb {R}$ a strictly monotone function. Prove that $f(A)$ is a set of first categories.

Let $f$ be differentiable on $[a, b]$ with bounded derivative. Prove that $f$ is of bounded variation on $[a, b]$.

Prove that the Dirichlet function is not of bounded variation.

Let $f$ be a function of bounded variation on $[a, b]$ and $[c, d] \subset [a, b]$. Prove that $f$ is of bounded variation on $[c, d]$.

Prove that every function of bounded variation is bounded.

Let $f, g:[a, b] \rightarrow \mathbb {R}$ be of bounded variation. Prove that so is $f g$.

Let $f:[a, b] \rightarrow \mathbb {R}$ be of bounded variation. Prove that so is $\left|f\right|$.

Let $f, g:[a, b] \rightarrow \mathbb {R}$ be of bounded variation and suppose that $1 / g$ is bounded on $[a, b]$. Prove that $f / g$ is of bounded variation.

Let $f:[a, b] \rightarrow \mathbb {R}$ and suppose that $f$ is of bounded variation on every closed subinterval of $(a, b)$. Prove or disprove: $f$ is of bounded variation on $[a, b]$.

Let

Determine whether $f$ is of bounded variation on $[0,1]$.

Let

Prove that $f$ is of bounded variation on $[0,1]$ if and only if $\alpha +\beta >0$ or both $\alpha +\beta =0$ and $\beta \geq 0$.

Let $g$ be a function of bounded variation on $[0,1]$ and let $f$ be a continuous function on $\mathbb {R}$. Prove or disprove: $f \circ g$ is of bounded variation on $[0,1]$.

Let $g$ be a function of bounded variation on $[0,1]$ and let $f$ be of bounded variation on $\mathbb {R}$. Prove or disprove: $f \circ g$ is of bounded variation on $[0,1]$.

Let $f$ be a function on $[0,1]$ that is not of bounded variation. Prove that there is a point $x_{0} \in [0,1]$ such that $f$ fails to be of bounded variation on each subinterval of $[0,1]$ that contains $x_{0}$ in its interior.

Give an example of a function that is differentiable on $[0,1]$ but not of bounded variation on $[0,1]$.

Let $f, g:[a, b] \rightarrow \mathbb {R}$. Prove that $V_{a}^{b}(f+g) \leq V_{a}^{b}(f)+V_{a}^{b}(g)$.

Let $f:[a, b] \rightarrow \mathbb {R}$ and let $\alpha \in \mathbb {R}$. Prove that $V_{a}^{b}(\alpha f)=\left|\alpha \right| V_{a}^{b}(f)$.

Let $f:[a, b] \rightarrow \mathbb {R}$. Prove that $V_{a}^{b}(f)=0$ if and only if $f$ is constant.

Let $f:[a, b] \rightarrow \mathbb {R}$ be a function of bounded variation and let $x_{0} \in (a, b)$. Prove that if $f$ is continuous from the left (right) at $x_{0}$, then so is $V_{a}^{x}(f)$.

For a function $f:[a, b] \rightarrow \mathbb {R}$ the length of the curve $y=f(x), a \leq x \leq b$ is defined by

where the supremum is taken over all partitions $P=\left\{ x_{0}, x_{1}, \ldots , x_{n}\right\}$ of $[a, b]$. Prove that $L(f)$ is finite if and only $f$ is of bounded variation on $[a, b]$.

Let $f$ be differentiable on $[a, b]$ with derivative $f^{\prime }$ Riemann integrable over $[a, b]$. Prove that

What if the integral of $\left|f^{\prime }\right|$ is a convergent improper integral?

Let $f$ be continuous on $[a, b]$ and suppose that $D^{+} f$ is Riemann integrable over $[a, b]$. Prove that

What if the integral of $\left|D^{+} f\right|$ is a convergent improper integral?

Let $f, g$ be differentiable on $[a, b]$ and let $C>0$ such that $\left|f^{\prime }\right| \leq C\left|g^{\prime }\right|$ on $[a, b]$. Suppose that either $g^{\prime }$ is Riemann integrable over $[a, b]$ or the integral of $\left|g^{\prime }\right|$ is a convergent improper integral. Prove that $f$ is of bounded variation on $[a, b]$.

In Exercises 9.3.23-9.3.26 find the total variation: $V_{0}^{50}\left(e^{x}\right)$.

In Exercises 9.3.23-9.3.26 find the total variation: $V_{1}^{2}(\ln x)$.

In Exercises 9.3.23-9.3.26 find the total variation: $V_{0}^{4 \pi }(\cos x)$.

In Exercises 9.3.23-9.3.26 find the total variation: $V_{-1}^{1}\left(x-x^{3}\right)$.

Let $f, g$ be continuous functions of bounded variation on $[0,1]$. Prove that the set $\left\{ (f(x), g(x)): x \in [0,1]\right\}$ cannot be the square $[0,1] \times [0,1]$. What if $f, g$ are merely continuous?

Let $\left\{ f_{n}\right\}$ be a sequence of real-valued functions on $[a, b]$ that converges pointwise on $[a, b]$ to the real-valued function $f$. Prove that

Find all functions $f$ of bounded variation on $[0,1]$ such that

Let $f$ be a function of bounded variation on a closed and bounded interval $[a, b]$. Prove that there exists a sequence of partitions $\left\{ P_{n}\right\}$ of $[a, b]$ for which the sequence $\left\{ V\left(f, P_{n}\right)\right\}$ is increasing and converges to $V_{a}^{b}(f)$.

Let $f:[a, b] \rightarrow \mathbb {R}$ be a continuous function of bounded variation and let $P=\left\{ x_{0}, x_{1}, \ldots , x_{n}\right\}$ be a partition of $[a, b]$. For $1 \leq k \leq n$, let $M_{k}$ and $m_{k}$ denote the maximum and the minimum, respectively, of $f$ on $\left[x_{k-1}, x_{k}\right]$, and $\Omega (f, P)=\sum_{k=1}^{n}\left(M_{k}-m_{k}\right)$. Prove that

Prove or disprove: There exists a continuous function that is not of bounded variation on any interval.

Let $f(x)=\sin x$ on $[0,2 \pi ]$. Find two increasing functions $h$ and $g$ for which $f=h-g$ on $[0,2 \pi ]$.