Differentiation and Integration

Show that there is a strictly increasing function $f:[0,1] \rightarrow \mathbb {R}$ that is continuous only at the irrational numbers in $[0,1]$.

Verify the assertion in the proof of the theorem that it suffices to only consider the case that $I$ is closed and bounded.

Let $E$ be any subset of $\mathbb {R}$ and $C$ a countable subset of $E$. Is there a monotone function on $E$ that is continuous only at points in $E \sim C$ ?

Show that the Vitali Covering Lemma does not extend to the case in which the covering collection has degenerate closed intervals.

Show that the Vitali Covering Lemma does extend to the case in which the covering collection comprises non-degenerate general intervals.

Is every continuous function $f:[0,1] \rightarrow \mathbb {R}$ the difference of monotone functions?

Let $I$ and $J$ be closed, bounded intervals and $\gamma > 0$ be such that $\ell (I) > \gamma \cdot \ell (J)$. Assume that $I \cap J \neq \emptyset$. Show that if $\gamma \geq 1 / 2$, then $J \subseteq 3 * I$. Is the same true if $0 < \gamma < 1 / 2$ ?

Show that $m(E) = 0$ 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 the $I_{k}$ 's and $\Sigma_{k = 1}^{\infty } \ell \left(I_{k}\right) < \infty$.

Let $E$ be a set of measure zero contained in the open interval ($a, b$). Let $\left\{ \left(c_{k}, d_{k}\right)\right\}_{k = 1}^{\infty }$ be a cover of $E$ as in the preceding problem. Define

Show that $f$ is increasing and fails to be differentiable at each point in $E$.

For real numbers $\alpha < \beta$ and $\gamma > 0$, show that if $g:[\alpha +\gamma , \beta +\gamma ] \rightarrow \mathbb {R}$ is integrable then

Prove this change of variables formula by successively considering simple functions, bounded, measurable functions, non-negative integrable functions, and general integrable functions. Use it to prove (7).

Compute the upper and lower derivatives of the characteristic function of the rationals.

Assume that $m^{*}(E) < \infty$ and let $\mathcal{F}$ be a collection of closed, bounded, non-degenerate intervals that cover $E$ in the sense of Vitali. Show that there is a countable, disjoint collection $\left\{ I_{k}\right\}_{k = 1}^{\infty }$ of intervals in $\mathcal{F}$ for which

Use the Vitali Covering Lemma to show that the union of any collection (countable or uncountable) of closed, bounded, non-degenerate intervals is measurable.

Define $f: \mathbb {R} \rightarrow \mathbb {R}$ by

Find the upper and lower derivatives of $f$ at $x = 0$.

For an integrable function $g:[a, b] \rightarrow \mathbb {R}$, define the antiderivative of $g$ to be the function $f$ defined on $[a, b]$ by

Show that $f$ is differentiable almost everywhere on $(a, b)$.

Verify (7).

Show that for $f:(a, b) \rightarrow \mathbb {R}$ and $c \in (a, b)$ a local minimizer for $f, \underline{D} f(c) \leq 0 \leq \bar{D} f(c)$.

For $f:[a, b] \rightarrow \mathbb {R}$ assume that $\underline{D} f \geq 0$ on $(a, b)$. Show that $f$ is increasing on $[a, b]$.

For functions $f:[a, b] \rightarrow \mathbb {R}$ and $g:[a, b] \rightarrow \mathbb {R}$, show that

Let $f$ be defined on $[a, b]$ and $g$ a continuous function on $[\alpha , \beta ]$ that is differentiable at $\gamma \in (\alpha , \beta )$ with $g(\gamma ) = c \in (a, b)$. Verify the following.

Show that a continuous function $f:[a, b] \rightarrow \mathbb {R}$ is Lipschitz if its upper and lower derivatives are bounded on $(a, b)$.

Show that for $f$ defined in the last example of this section, $f^{\prime }:[0,1] \rightarrow \mathbb {R}$ is not integrable.

Let $f$ be the Dirichlet function, the characteristic function of the rationals in $[0,1]$. Is $f:[0,1] \rightarrow \mathbb {R}$ of bounded variation?

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

Let $f:[a, b] \rightarrow \mathbb {R}$ be a step-function. Find a formula for its total variation.

Suppose that $f:[0,1] \rightarrow \mathbb {R}$ is continuous. Must there be a non-degenerate, closed subinterval $[a, b]$ of $[0,1]$ for which the restriction of $f$ to $[a, b]$ is of bounded variation?

Let $P$ be a partition of $[a, b]$ that is a refinement of the partition $P^{\prime }$. For $f:[a, b] \rightarrow \mathbb {R}$, show that $V\left(f, P^{\prime }\right) \leq V(f, P)$.

Assume that $f:[a, b] \rightarrow \mathbb {R}$ is of bounded variation. Show that there is 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 $T V(f)$.

Let the sequence $\left\{ f_{n}:[a, b] \rightarrow R\right\}$ converge pointwise to $f:[a, b] \rightarrow \mathbb {R}$. Show that

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

For $\alpha$ and $\beta$ positive numbers, define the function $f$ on $[0,1]$ by

Show that if $\alpha > \beta$, then $f:[0,1] \rightarrow \mathbb {R}$ is of bounded variation, by showing that $f^{\prime }$ is integrable over $[0,1]$. Then show that if $\alpha \leq \beta$, then $f:[0,1] \rightarrow \mathbb {R}$ is not of bounded variation.

Let $f:[0,1] \rightarrow \mathbb {R}$ fail to be of bounded variation. Show that there is a point $x_{0}$ in $[0,1]$ such that there are subintervals of $[0,1]$ that contain $x_{0}$ and have arbitrarily small length on which $f$ fails to be of bounded variation.

Let $f:[0,1] \rightarrow \mathbb {R}$ be continuous, and be absolutely continuous on $[\epsilon , 1]$ for each $0 < \epsilon < 1$.

Verify that the inclusions in (19) are strict.

Show that a Lipschitz function $f:[a, b] \rightarrow \mathbb {R}$ is the difference of increasing Lipschitz functions.

Assume that $f:[a, b] \rightarrow \mathbb {R}$ is increasing. Show that $f$ is absolutely continuous if and only if for each $\epsilon > 0$, there is a $\delta > 0$ such that for every countable, disjoint collection $\left\{ \left(a_{k}, b_{k}\right)\right\}_{k = 1}^{\infty }$ of open intervals in ($a, b$),

Show that if $f:[a, b] \rightarrow \mathbb {R}$ is increasing and absolutely continuous, then it maps sets of measure zero to sets of measure zero.

Let $f:[a, b] \rightarrow \mathbb {R}$ be continuous. Conclude from the continuity of $f$ and the compactness of $[a, b]$ that $f$ maps closed sets to closed sets and therefore maps $F_{\sigma }$ sets to $F_{\sigma }$ sets.

Show that if $f:[a, b] \rightarrow \mathbb {R}$ is increasing and absolutely continuous, then $f$ maps measurable sets to measurable sets.

Show that if $f:[a, b] \rightarrow \mathbb {R}$ is continuous and increasing, then $f$ is absolutely continuous if and only if for each $\epsilon$, there is a $\delta > 0$ such that for each measurable subset $E$ of $[a, b]$,

Show that the Cantor-Lebesgue function $\varphi$ is not absolutely continuous on $[0,1]$ by arguing that the function $\psi$, defined by $\psi (x) = x+\varphi (x)$ for $0 \leq x \leq 1$, maps the Cantor, a set of measure zero, to a set of measure 1.

Show that both the sum and product of absolutely continuous functions are absolutely continuous.

Define the functions $f$ and $g$ on $[-1,1]$ by $f(x) = x^{\frac{1}{3}}$ for $-1 \leq x \leq 1$ and

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be Lipschitz and $g:[a, b] \rightarrow \mathbb {R}$ be absolutely continuous. Show that the composition $f \circ g:[a, b] \rightarrow \mathbb {R}$ is absolutely continuous.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be absolutely continuous and $g:[a, b] \rightarrow \mathbb {R}$ be absolutely continuous and strictly increasing. Show that the composition $f \circ g:[a, b] \rightarrow \mathbb {R}$ is absolutely continuous.

Prove the converse of the implication in Theorem 11.

Theorem 11: If the function $f: \left[a, b+1\right] \rightarrow \mathbb {R}$ is absolutely continuous, then the collection of divided difference functions $\mathcal{F} = \left\{ \operatorname {Diff}_{h} f: \left[a, b\right] \rightarrow \mathbb {R}\right\}_{0 < h \leq 1}$ is uniformly integrable.

The Cantor-Lebesgue function $\varphi$ is continuous and increasing on $[0,1]$. Conclude from Theorem 12 that $\varphi$ is not absolutely continuous on $[0,1]$.

Theorem 12: If the function $f: \left[a, b\right] \rightarrow \mathbb {R}$ is absolutely continuous, then it is differentiable almost everywhere on $(a, b)$, its derivative $f'$ is integrable, and $\int_{a}^{b} f' \, dm = f(b) - f(a)$. Compare this reasoning with that proposed in Problem 42.

Let $f:[a, b] \rightarrow \mathbb {R}$ be continuous and differentiable almost everywhere on ($a, b$). Show that

Let $f:[a, b] \rightarrow \mathbb {R}$ be continuous and differentiable almost everywhere on ($a, b$). Show that if $\left\{ \operatorname {Diff}_{1 / n} f\right\}$ is uniformly integrable over $[a, b]$, then

Let $f:[a, b] \rightarrow \mathbb {R}$ be continuous and differentiable almost everywhere on ($a, b$). Suppose that there is a non-negative integrable function $g:[a, b] \rightarrow \mathbb {R}$ for which

Show that

If $f:[a, b] \rightarrow \mathbb {R}$ is absolutely continuous, show that $f$ is Lipschitz if and only if there is a $c \geq 0$ for which $\left|f^{\prime }\right| \leq c$ almost everywhere on $[a, b]$.

Let $f:[a, b] \rightarrow \mathbb {R}$ be of bounded variation, and define $v(x) = T V\left(f_{[a, x]}\right)$ for all $x \in [a, b]$.

Construct an absolutely continuous, strictly increasing function $f:[0,1] \rightarrow \mathbb {R}$ for which $f^{\prime } = 0$ on a set of positive measure.

For $f$ and $g$ as in the statement of Theorem 17, is it possible to establish the change of variables formula (30) by directly verifying that

Theorem 17 (Change of Variables): Let the function $f: \left[a, b\right] \rightarrow \mathbb {R}$ be integrable. If the function $g: \left[c, d\right] \rightarrow \mathbb {R}$ is increasing and absolutely continuous, $a = g(c)$ and $b = g(d)$, then $\int_{a}^{b} f \, dm = \int_{c}^{d} \left[f \circ g\right] \cdot g' \, dm$ — the assertion (30) that this problem asks whether one can verify directly via the equation above.

Let $f:[a, b] \rightarrow \mathbb {R}$ preserve sets of measure zero. Show that the composition $g \circ f$ is measurable whenever $g: \mathbb {R} \rightarrow \mathbb {R}$ is measurable if this holds for simple functions $g$.

Let $f:[a, b] \rightarrow \mathbb {R}$ be an increasing, absolutely continuous function. Prove that for each measurable subset $E$ of $[a, b]$,

Find an increasing function $f:[a, b] \rightarrow \mathbb {R}$ that preserves measurable sets for which inequality (35) does not hold.

Prove that if $f:[a, b] \rightarrow \mathbb {R}$ is not absolutely continuous, then there is an integrable (not just measurable) function $g: \mathbb {R} \rightarrow \mathbb {R}$ for which $g \circ f$ is not measurable.

Show that $\varphi :(a, b) \rightarrow \mathbb {R}$ is convex if and only if for points $x_{1}, \ldots , x_{n}$ in ($a, b$) and nonnegative numbers $\lambda_{1}, \ldots , \lambda_{n}$ such that $\sum_{k = 1}^{n} \lambda_{k} = 1$,

Use this to directly prove Jensen's Inequality for $f$ a simple function.

Show that a continuous function $\varphi :(a, b) \rightarrow \mathbb {R}$ is convex if and only if

A function $\varphi : I \rightarrow \mathbb {R}$ on a general interval $I$ is said to be convex provided that it is continuous and (38) holds for all $x_{1}, x_{2} \in I$. Is a convex function on a closed, bounded interval $[a, b]$ necessarily Lipschitz on $[a, b]$ ?

Let $\varphi :(a, b) \rightarrow \mathbb {R}$ have a second derivative at each point in $(a, b)$. Show that $\varphi$ is convex if and only if $\varphi^{\prime \prime }$ is non-negative.

Let $a \geq 0$ and $b > 0$. Show that the function $\varphi (t) = (a+b t)^{p}$ is convex on $[0, \infty )$ for $1 \leq p < \infty$.

For what functions $\varphi$ is Jensen's Inequality always an equality?

State and prove a version of Jensen's Inequality on a general closed, bounded interval $[a, b]$.

Let $f:[0,1] \rightarrow \mathbb {R}$ be integrable. Show that

Let $\left\{ \alpha_{n}\right\}$ be a sequence of non-negative numbers whose sum is 1 and $\left\{ \zeta_{n}\right\}$ a sequence of positive numbers. Show that

Let $g:[0,1] \rightarrow \mathbb {R}$ be positive and measurable. Show that $\log \left(\int_{0}^{1} g(x) d x\right) \geq \int_{0}^{1} \log (g(x)) d x$ whenever each side is defined.

Let $\varphi : \mathbb {R} \rightarrow \mathbb {R}$ be continuous. Show that if there are $c_{1} \geq 0, c_{2} \geq 0$ for which (43) holds, then $\varphi \circ f:[0,1] \rightarrow \mathbb {R}$ is integrable if $f:[0,1] \rightarrow \mathbb {R}$ is integrable.