6.3.E: Problems on Differentiable Functions
Complete the missing details in the proofs of this section.
Verify Note 1. Describe \(\left[f^{\prime}(\vec{p})\right]\) for \(f : E^{1} \rightarrow E^{m},\) too. Give examples.
\(\Rightarrow\) \(A \operatorname{map} f : E^{\prime} \rightarrow E\) 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
\[\lim _{\vec{x} \rightarrow \vec{p}} \frac{1}{|\Delta \vec{x}|}|\Delta f||\Delta g|=0.\]
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.]
\(\Rightarrow\) 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 \(f(\vec{p})=f(\vec{q})\).]
Do Problem 5 in case \(G\) is any open polygon-connected set in \(E^{\prime}.\) (See Chapter 4, §9.)
\(\Rightarrow\) 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
\[\nabla h(\vec{p})=a \nabla f(\vec{p})+b \nabla g(\vec{p}).\]
\(\Rightarrow\) 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 \(g F.\)]
\(\Rightarrow\) 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, \(d f=d f_{re} + i \cdot d f_{im}.\)
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.
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 \(\vec{p},\) and \(f(\vec{p})=0.\) Find \(d h.\)
Do Problem 8(i) for dot products \(h=f \cdot g\) of functions \(f, g : E^{\prime} \rightarrow E^{m}\) \((C^{m}).\)
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).]
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.]