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# 6.3.E: Problems on Differentiable Functions

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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.]