6.3.E: Problems on Differentiable Functions

Exercise $\PageIndex{1}$

Complete the missing details in the proofs of this section.

Exercise $\PageIndex{2}$

Verify Note 1. Describe $\left[f^{\prime}(\vec{p})\right]$ for $f : E^{1} \rightarrow E^{m},$ too. Give examples.

Exercise $\PageIndex{3}$

$\Rightarrow$ $A \operatorname{map} f : E^{\prime} \rightarrow E$ is said to satisfy a Lipschitz condition $(L)$ of order $\alpha>0$ at $\vec{p}$ iff
$$(\exists \delta>0)\left(\exists K \in E^{1}\right)\left(\forall \vec{x} \in G_{\neg \vec{p}}(\delta)\right) \quad|f(\vec{x})-f(\vec{p})| \leq K|\vec{x}-\vec{p}|^{\alpha}.$$
Prove the following.
(i) This implies continuity at $\vec{p}$ (but not conversely; see Problem 7 in Chapter 5, §1).
(ii) $L$ of order $>1$ implies differentiability at $\vec{p},$ with $d f(\vec{p} ; \cdot)=0$ on $E^{\prime}.$
(iii) Differentiability at $\vec{p}$ implies $L$ of order 1 (apply Theorem 1 in §2 to $\phi=d f$).
(iv) If $f$ and $g$ are differentiable at $\vec{p},$ then
$$\lim _{\vec{x} \rightarrow \vec{p}} \frac{1}{|\Delta \vec{x}|}|\Delta f||\Delta g|=0.$$

Exercise $\PageIndex{4}$

For the functions of Problem 5 in §1, find those $\vec{p}$ at which $f$ is differentiable. Find
$$\nabla f(\vec{p}), d f(\vec{p} ; \cdot), \text { and }\left[f^{\prime}(\vec{p})\right].$$
[Hint: Use Theorem 3 and Corollary 1.]

Exercise $\PageIndex{5}$

$\Rightarrow$ Prove the following statements.
(i) If $f : E^{\prime} \rightarrow E$ is constant on an open globe $G \subset E^{\prime},$ it is differentiable at each $\vec{p} \in G,$ and $d f(\vec{p}, \cdot)=0$ on $E^{\prime}.$
(ii) If the latter holds for each $\vec{p} \in G-Q$ ($Q$ countable), then $f$ is constant on $G$ (even on $\overline{G}$) provided $f$ is relatively continuous there.
[Hint: Given $\vec{p}, \vec{q} \in G,$ use Theorem 2 in §1 to get $f(\vec{p})=f(\vec{q})$.]

Exercise $\PageIndex{6}$

Do Problem 5 in case $G$ is any open polygon-connected set in $E^{\prime}.$ (See Chapter 4, §9.)

Exercise $\PageIndex{7}$

$\Rightarrow$ Prove the following.
(i) If $f, g : E^{\prime} \rightarrow E$ are differentiable at $\vec{p},$ so is
$$h=a f+b g,$$
for any scalars $a, b$ (if $f$ and $g$ are scalar valued, $a$ and $b$ may be vectors; moreover,
$$d(a f+b g)=a d f+b d g,$$
i.e.,
$$d h(\vec{p} ; \vec{t})=a d f(\vec{p} ; \vec{t})+b d g(\vec{p} ; \vec{t}), \quad \vec{t} \in E^{\prime}.$$
(ii) In case $f, g : E^{m} \rightarrow E^{1}$ or $C^{m} \rightarrow C,$ deduce also that
$$\nabla h(\vec{p})=a \nabla f(\vec{p})+b \nabla g(\vec{p}).$$

Exercise $\PageIndex{8}$

$\Rightarrow$ Prove that if $f, g : E^{\prime} \rightarrow E^{1}(C)$ are differentiable at $\vec{p},$ then so are
$$h=g f \text { and } k=\frac{g}{f}.$$
(the latter, if $f(\vec{p}) \neq 0).$ Moreover, with $a=f(\vec{p})$ and $b=g(\vec{p}),$ show that
(i) $d h=a d g+b d f$ and
(ii) $d k=(a d g-b d f) / a^{2}$.
If further $E^{\prime}=E^{n}\left(C^{n}\right),$ verify that
(iii) $\nabla h(\vec{p})=a \nabla g(\vec{p})+b \nabla f(\vec{p})$ and
(iv) $\nabla k(\vec{p})=(a \nabla g(\vec{p})-b \nabla f(\vec{p})) / a^{2}$.
Prove (i) and (ii) for vector-valued $g,$ too.
[Hints: (i) Set $\phi=a d g+b d f,$ with $a$ and $b$ as above. Verify that
$$\Delta h-\phi(\vec{t})=g(\vec{p})(\Delta f-d f(\vec{t}))+f(\vec{p})(\Delta g-d g(\vec{t}))+(\Delta f)(\Delta g).$$
Use Problem 3(iv) and Definition 1.
(ii) Let $F(\vec{t})=1 / f(\vec{t}).$ Show that $d F=-d f / a^{2}.$ Then apply (i) to $g F.$]

Exercise $\PageIndex{9}$

$\Rightarrow$ Let $f : E^{\prime} \rightarrow E^{m}\left(C^{m}\right), f=\left(f_{1}, \ldots, f_{m}\right).$ Prove that
(i) $f$ is linear iff all its $m$ components $f_{k}$ are;
(ii) $f$ is differentiable at $\vec{p}$ iff all $f_{k}$ are, and then $d f=\left(d f_{1}, \ldots, d f_{m}\right)$. Hence if $f$ is complex, $d f=d f_{re} + i \cdot d f_{im}.$

Exercise $\PageIndex{10}$

Prove the following statements.
(i) If $f \in L\left(E^{\prime}, E\right)$ then $f$ is differentiable on $E^{\prime},$ and $d f(\vec{p} ; \cdot)=f$, $\vec{p} \in E^{\prime}.$
(ii) Such is any first-degree monomial, hence any sum of such monomials.

Exercise $\PageIndex{11}$

Any rational function is differentiable in its domain.
[Hint: Use Problems 10(i), 7, and 8. Proceed as in Theorem 3 in Chapter 4, §3.]

Exercise $\PageIndex{12}$

Do Problem 8(i) in case $g$ is only continuous at $\vec{p},$ and $f(\vec{p})=0.$ Find $d h.$

Exercise $\PageIndex{13}$

Do Problem 8(i) for dot products $h=f \cdot g$ of functions $f, g : E^{\prime} \rightarrow E^{m}$ $(C^{m}).$

Exercise $\PageIndex{14}$

Prove the following.
(i) If $\phi \in L\left(E^{n}, E^{1}\right)$ or $\phi \in L\left(C^{n}, C\right),$ then $\|\phi\|=|\vec{v}|,$ with $\vec{v}$ as in §2, Theorem 2(ii).
(ii) If $f : E^{n} \rightarrow E^{1}\left(f : C^{n} \rightarrow C^{1}\right)$ is differentiable at $\vec{p},$ then
$$\|d f(\vec{p} ; \cdot)\|=|\nabla f(\vec{p})|.$$
Moreover, in case $f : E^{n} \rightarrow E^{1}$,
$$|\nabla f(\vec{p})| \geq D_{\vec{u}} f(\vec{p}) \quad \text {if }|\vec{u}|=1$$
and
$$|\nabla f(\vec{p})|=D_{\vec{u}} f(\vec{p}) \quad \text {when } \vec{u}=\frac{\nabla f(\vec{p})}{|\nabla f(\vec{p})|;}$$
thus
$$|\nabla f(\vec{p})|=\max _{|\vec{u}|=1} D_{\vec{u}} f(\vec{p}).$$
[Hints: Use the equality case in Theorem 4(c') of Chapter 3, §§1-3. Use formula (7), Corollary 2, and Theorem 2(ii).]

Exercise $\PageIndex{15}$

Show that Theorem 3 holds even if
(i) $D_{1} f$ is discontinuous at $\vec{p},$ and
(ii) $f$ has partials on $A-Q$ only $(Q$ countable, $\vec{p} \notin Q),$ provided $f$ is continuous on $A$ in each of the last $n-1$ variables.
[Hint: For $k=1,$ formula (13) still results by definition of $D_{1} f,$ if a suitable $\delta$ has been chosen.]

Exercise $\PageIndex{16*}$

Show that Theorem 3 and Problem 15 apply also to any $f : E^{\prime} \rightarrow E$ where $E^{\prime}$ is $n$-dimensional with basis $\left\{\vec{u}_{1}, \ldots, \vec{u}_{n}\right\}$ (see Problem 12 in §2) if we write $D_{k} f$ for $D_{\vec{u}_{k}} f$.
[Hints: Assume $\left|\vec{u}_{k}\right|=1,1 \leq k \leq n$ (if not, replace $\vec{u}_{k}$ by $\vec{u}_{k} /\left|\vec{u}_{k}\right|;$ show that this yields another basis). Modify the proof so that the $\vec{p}_{k}$ are still in $G_{\vec{p}}(\delta).$ Caution: The standard norm of $E^{n}$ does not apply here.]

Exercise $\PageIndex{17}$

Let $f_{k} : E^{1} \rightarrow E^{1}$ be differentiable at $p_{k} (k=1, \ldots, n).$ For $\vec{x}=(x_{1}, \ldots, x_{n}) \in E^{n},$ set
$$F(\vec{x})=\sum_{k=1}^{n} f_{k}\left(x_{k}\right) \text { and } G(\vec{x})=\prod_{k=1}^{n} f_{k}\left(x_{k}\right).$$
Show that $F$ and $G$ are differentiable at $\vec{p}=\left(p_{1}, \ldots, p_{n}\right).$ Express $\nabla F(\vec{p})$ and $\nabla G(\vec{p})$ in terms of the $f_{k}^{\prime}\left(p_{k}\right)$.
[Hint: In order to use Problems 7 and 8, replace the $f_{k}$ by suitable functions defined on $E^{n}.$ For $\nabla G(\vec{p}),$ "imitate" Problem 6 in Chapter 5, §1.]