Math Problems Vault

Problem 6.48

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.

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Problem 6.49

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

\begin{gathered} \int _{a}^{b} f^{\prime } d m = f(b)-f(a) \\ \text{ if and only if } \\ \int _{a}^{b}\left[\lim _{n \rightarrow \infty } \operatorname {Diff}_{1 / n} f\right] d m = \lim _{n \rightarrow \infty }\left[\int _{a}^{b} \operatorname {Diff}_{1 / n} f\right] d m \end{gathered}

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Problem 6.50

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

\int _{a}^{b} f^{\prime } d m = f(b)-f(a)

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Problem 6.51

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

\left|\operatorname {Diff}_{1 / n} f\right| \leq g \text{ on }[a, b] \text{ for all } n .

Show that

\int _{a}^{b} f^{\prime } d m = f(b)-f(a)

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Problem 6.52

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 $[a, b]$ .

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Problem 6.53

(i)(ii)(iii)

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

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(i)

Show that $\left|f^{\prime }\right| \leq v^{\prime }$ almost everywhere on $[a, b]$ , and deduce from this that

\int _{a}^{b}\left|f^{\prime }\right| d m \leq T V(f)

(ii)

Show that the above is an equality if and only if $f$ is absolutely continuous on $[a, b]$ .

(iii)

Compare parts (i) and (ii) with Corollaries 4 and 12, respectively.

Problem 6.54

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

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Problem 6.55

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

\frac{d}{d x}\left[\int _{a}^{g(x)} f d m-\int _{c}^{x}[f \circ g] \cdot g^{\prime } d m\right] = 0 \text{ for almost all } x \in (c, d) ?

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.

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