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6.3.E: Problems on Differentiable Functions

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Exercise 6.3.E.1

Complete the missing details in the proofs of this section.

Exercise 6.3.E.2

Verify Note 1. Describe [f(p)] for f:E1Em, too. Give examples.

Exercise 6.3.E.3

Amapf:EE is said to satisfy a Lipschitz condition (L) of order α>0 at p iff
(δ>0)(KE1)(xG¬p(δ))|f(x)f(p)|K|xp|α.
Prove the following.
(i) This implies continuity at p (but not conversely; see Problem 7 in Chapter 5, §1).
(ii) L of order >1 implies differentiability at p, with df(p;)=0 on E.
(iii) Differentiability at p implies L of order 1 (apply Theorem 1 in §2 to ϕ=df).
(iv) If f and g are differentiable at p, then
limxp1|Δx||Δf||Δg|=0.

Exercise 6.3.E.4

For the functions of Problem 5 in §1, find those p at which f is differentiable. Find
f(p),df(p;), and [f(p)].
[Hint: Use Theorem 3 and Corollary 1.]

Exercise 6.3.E.5

Prove the following statements.
(i) If f:EE is constant on an open globe GE, it is differentiable at each pG, and df(p,)=0 on E.
(ii) If the latter holds for each pGQ (Q countable), then f is constant on G (even on ¯G) provided f is relatively continuous there.
[Hint: Given p,qG, use Theorem 2 in §1 to get f(p)=f(q).]

Exercise 6.3.E.6

Do Problem 5 in case G is any open polygon-connected set in E. (See Chapter 4, §9.)

Exercise 6.3.E.7

Prove the following.
(i) If f,g:EE are differentiable at p, so is
h=af+bg,
for any scalars a,b (if f and g are scalar valued, a and b may be vectors; moreover,
d(af+bg)=adf+bdg,
i.e.,
dh(p;t)=adf(p;t)+bdg(p;t),tE.
(ii) In case f,g:EmE1 or CmC, deduce also that
h(p)=af(p)+bg(p).

Exercise 6.3.E.8

Prove that if f,g:EE1(C) are differentiable at p, then so are
h=gf and k=gf.
(the latter, if f(p)0). Moreover, with a=f(p) and b=g(p), show that
(i) dh=adg+bdf and
(ii) dk=(adgbdf)/a2.
If further E=En(Cn), verify that
(iii) h(p)=ag(p)+bf(p) and
(iv) k(p)=(ag(p)bf(p))/a2.
Prove (i) and (ii) for vector-valued g, too.
[Hints: (i) Set ϕ=adg+bdf, with a and b as above. Verify that
Δhϕ(t)=g(p)(Δfdf(t))+f(p)(Δgdg(t))+(Δf)(Δg).
Use Problem 3(iv) and Definition 1.
(ii) Let F(t)=1/f(t). Show that dF=df/a2. Then apply (i) to gF.]

Exercise 6.3.E.9

Let f:EEm(Cm),f=(f1,,fm). Prove that
(i) f is linear iff all its m components fk are;
(ii) f is differentiable at p iff all fk are, and then df=(df1,,dfm). Hence if f is complex, df=dfre+idfim.

Exercise 6.3.E.10

Prove the following statements.
(i) If fL(E,E) then f is differentiable on E, and df(p;)=f, pE.
(ii) Such is any first-degree monomial, hence any sum of such monomials.

Exercise 6.3.E.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 6.3.E.12

Do Problem 8(i) in case g is only continuous at p, and f(p)=0. Find dh.

Exercise 6.3.E.13

Do Problem 8(i) for dot products h=fg of functions f,g:EEm (Cm).

Exercise 6.3.E.14

Prove the following.
(i) If ϕL(En,E1) or ϕL(Cn,C), then 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.]


6.3.E: Problems on Differentiable Functions is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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