6.3.E: Problems on Differentiable Functions
( \newcommand{\kernel}{\mathrm{null}\,}\)
Complete the missing details in the proofs of this section.
Verify Note 1. Describe [f′(→p)] for f:E1→Em, too. Give examples.
⇒ Amapf:E′→E is said to satisfy a Lipschitz condition (L) of order α>0 at →p iff
(∃δ>0)(∃K∈E1)(∀→x∈G¬→p(δ))|f(→x)−f(→p)|≤K|→x−→p|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
lim→x→→p1|Δ→x||Δf||Δg|=0.
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.]
⇒ Prove the following statements.
(i) If f:E′→E is constant on an open globe G⊂E′, it is differentiable at each →p∈G, and df(→p,⋅)=0 on E′.
(ii) If the latter holds for each →p∈G−Q (Q countable), then f is constant on G (even on ¯G) provided f is relatively continuous there.
[Hint: Given →p,→q∈G, use Theorem 2 in §1 to get f(→p)=f(→q).]
Do Problem 5 in case G is any open polygon-connected set in E′. (See Chapter 4, §9.)
⇒ Prove the following.
(i) If f,g:E′→E 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),→t∈E′.
(ii) In case f,g:Em→E1 or Cm→C, deduce also that
∇h(→p)=a∇f(→p)+b∇g(→p).
⇒ Prove that if f,g:E′→E1(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=(adg−bdf)/a2.
If further E′=En(Cn), verify that
(iii) ∇h(→p)=a∇g(→p)+b∇f(→p) and
(iv) ∇k(→p)=(a∇g(→p)−b∇f(→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
Δh−ϕ(→t)=g(→p)(Δf−df(→t))+f(→p)(Δg−dg(→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.]
⇒ Let f:E′→Em(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+i⋅dfim.
Prove the following statements.
(i) If f∈L(E′,E) then f is differentiable on E′, and df(→p;⋅)=f, →p∈E′.
(ii) Such is any first-degree monomial, hence any sum of such monomials.
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.]
Do Problem 8(i) in case g is only continuous at →p, and f(→p)=0. Find dh.
Do Problem 8(i) for dot products h=f⋅g of functions f,g:E′→Em (Cm).
Prove the following.
(i) If ϕ∈L(En,E1) or ϕ∈L(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).]
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.]
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.]
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.]