Help with D&K Lemma 2.2.7 (Hadamard) Proof: Qs on Continuity of \phi_a

In summary, Duistermaat and Kolk's Lemma 2.2.7 and its proof show that if $f$ is differentiable at a, then $\phi_a$ is continuous at a.
  • #1
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I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 2: Differentiation ... ...

I need help with another aspect of the proof of Lemma 2.2.7 (Hadamard...) ... ...

Duistermaat and Kolk's Lemma 2.2.7 and its proof read as follows:https://www.physicsforums.com/attachments/7837
https://www.physicsforums.com/attachments/7838
Near to the end of the above text D&K write the following:

" ... ... A direct computation gives \(\displaystyle \| \epsilon_a(h) h^t \|_{ Eucl } = \| \epsilon_a(h) \| \| h \|\), hence\(\displaystyle \lim_{ h \rightarrow 0 } \frac{ \| \epsilon_a(h) h^t \|_{ Eucl } }{ \| h \|^2 } = \lim_{ h \rightarrow 0 } \frac{ \| \epsilon_a(h) \| }{ \| h \| } = 0 \)This shows that \(\displaystyle \phi_a\) is continuous at \(\displaystyle a\). ... ... "

My questions are as follows:

Question 1

... how/why does the above show that \(\displaystyle \phi_a\) is continuous at \(\displaystyle a\). ... ...?

Can someone please demonstrate explicitly, formally and rigorously that \(\displaystyle \phi_a\) is continuous at \(\displaystyle a\). ... ...?Question 2

How/why does the proof of Hadamard's Lemma 2.2.7 imply that \(\displaystyle f\) is continuous at \(\displaystyle a\) if \(\displaystyle f\) is differentiable at \(\displaystyle a\) ... ?
Help will be much appreciated ... ...

Peter==========================================================================================

NOTE:

The start of D&K's section on differentiable mappings may help readers of the above post understand the context and notation of the post ... so I am providing the same as follows:
View attachment 7839
View attachment 7840
The start of D&K's section on linear mappings may also help readers of the above post understand the context and notation of the post ... so I am providing the same as follows:

View attachment 7841
View attachment 7842
View attachment 7843
Hope the above helps readers understand the context and notation of the post ...

Peter
 
Last edited:
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  • #2
Hi, Peter.

Peter said:
Question 1

... how/why does the above show that \(\displaystyle \phi_a\) is continuous at \(\displaystyle a\). ... ...?

Can someone please demonstrate explicitly, formally and rigorously that \(\displaystyle \phi_a\) is continuous at \(\displaystyle a\). ... ...?

This question is perhaps a little deeper than it seems initially. The subtlety lies in the fact that $\phi_{a}(x)$ is an operator-valued function; i.e., for each $x$, $\phi_{a}(x)$ is a linear operator.

Step 1: Understanding Continuity
When the authors say "$\phi_{a}$ is continuous at $a$," what they mean is the operators $\phi_{a}(x)$ and $\phi_{a}(a)$ can be made arbitrarily close for $x$ sufficiently close to $a$ (i.e., the "distance" from $\phi_{a}(x)$ to $\phi_{a}(a)$ vanishes in the limit as $x\rightarrow a$). Symbolically what we want to show is that
$$\lim_{x\rightarrow a}\phi_{a}(x)=\phi_{a}(a)$$
Note that, at least symbolically, this is exactly the same notion as continuity in the case of a real-valued function of a single variable; i.e., $f$ is continuous at $a$ means
$$\lim_{x\rightarrow a}f(x)=f(a).$$
Step 2: Understanding Operator Norms
This begs the question: How do we measure the distance between operators? Depending on the context, there are several ways one can define a norm on a space of functions (this is not unlike your previous posts where you asked about different norms on $\mathbb{R}^{n}$). In the case of your post, the authors are (implicitly) using a notion known as "strong (operator) continuity," where, by definition,
$$\lim_{x\rightarrow a}\phi_{a}(x)=\phi_{a}(a)$$
provided
$$\lim_{x\rightarrow a}\|\phi_{a}(x)y-\phi_{a}(a)y\|=0,\qquad\forall y\in\mathbb{R}^{n},$$
where the norm above is the norm in $\mathbb{R}^{p}$ (which makes sense because $\phi_{a}(x)$ and $\phi_{a}(a)$ are operators taking elements of $\mathbb{R}^{n}$ into $\mathbb{R}^{p}$). Note that this condition which defines the meaning of the limit is the reason the authors need to introduce a $y$ in the equation
$$\phi_{a}(x)y=Df(a)y+\cdots$$
Step 3: Answering Your Question
From what was discussed in Step 2, we must show
$$\lim_{x\rightarrow a}\|\phi_{a}(x)y-\phi_{a}(a)y\|=0,\qquad\forall y\in\mathbb{R}^{n}.$$
Note that, by the authors' definition of $\phi_{a}(x)$, $\phi_{a}(a)=Df(a)$. Looking at the equation
$$\phi_{a}(x)y=Df(a)y+\cdots,$$
we see $\phi_{a}(x)-\phi_{a}(a)=\phi_{a}(x)-Df(a)=\cdots,$ so when the authors show
$$\lim_{h\rightarrow 0}\frac{\|\epsilon_{a}(h)h^{t}\|}{\|h\|^{2}}=0,$$
they are really proving
$$\lim_{x\rightarrow a}\|\phi_{a}(x)y-\phi_{a}(a)y\|=0,\qquad\forall y\in\mathbb{R}^{n},$$
which is precisely what we want to establish when proving $\phi_{a}(x)$ is continuous at $a$.

Peter said:
Question 2
How/why does the proof of Hadamard's Lemma 2.2.7 imply that \(\displaystyle f\) is continuous at \(\displaystyle a\) if \(\displaystyle f\) is differentiable at \(\displaystyle a\) ... ?

To prove that $f$ is continuous at $a$, we want to show that
$$\lim_{x\rightarrow a}f(x)=f(a),$$
where we note that $f$ is a function, not an operator, so we don't need to introduce $y$ as we did previously. From the above we know
$$\lim_{x\rightarrow a}\phi_{a}(x)=\phi_{a}(a)=Df(a),$$
so
$$\lim_{x\rightarrow a} f(x)=\lim_{x\rightarrow a} f(a)+\lim_{x\rightarrow a}\phi_{a}(x)(x-a)=f(a)+Df(a)(a-a)=f(a),$$
as desired.
 
  • #3
GJA said:
Hi, Peter.
This question is perhaps a little deeper than it seems initially. The subtlety lies in the fact that $\phi_{a}(x)$ is an operator-valued function; i.e., for each $x$, $\phi_{a}(x)$ is a linear operator.

Step 1: Understanding Continuity
When the authors say "$\phi_{a}$ is continuous at $a$," what they mean is the operators $\phi_{a}(x)$ and $\phi_{a}(a)$ can be made arbitrarily close for $x$ sufficiently close to $a$ (i.e., the "distance" from $\phi_{a}(x)$ to $\phi_{a}(a)$ vanishes in the limit as $x\rightarrow a$). Symbolically what we want to show is that
$$\lim_{x\rightarrow a}\phi_{a}(x)=\phi_{a}(a)$$
Note that, at least symbolically, this is exactly the same notion as continuity in the case of a real-valued function of a single variable; i.e., $f$ is continuous at $a$ means
$$\lim_{x\rightarrow a}f(x)=f(a).$$
Step 2: Understanding Operator Norms
This begs the question: How do we measure the distance between operators? Depending on the context, there are several ways one can define a norm on a space of functions (this is not unlike your previous posts where you asked about different norms on $\mathbb{R}^{n}$). In the case of your post, the authors are (implicitly) using a notion known as "strong (operator) continuity," where, by definition,
$$\lim_{x\rightarrow a}\phi_{a}(x)=\phi_{a}(a)$$
provided
$$\lim_{x\rightarrow a}\|\phi_{a}(x)y-\phi_{a}(a)y\|=0,\qquad\forall y\in\mathbb{R}^{n},$$
where the norm above is the norm in $\mathbb{R}^{p}$ (which makes sense because $\phi_{a}(x)$ and $\phi_{a}(a)$ are operators taking elements of $\mathbb{R}^{n}$ into $\mathbb{R}^{p}$). Note that this condition which defines the meaning of the limit is the reason the authors need to introduce a $y$ in the equation
$$\phi_{a}(x)y=Df(a)y+\cdots$$
Step 3: Answering Your Question
From what was discussed in Step 2, we must show
$$\lim_{x\rightarrow a}\|\phi_{a}(x)y-\phi_{a}(a)y\|=0,\qquad\forall y\in\mathbb{R}^{n}.$$
Note that, by the authors' definition of $\phi_{a}(x)$, $\phi_{a}(a)=Df(a)$. Looking at the equation
$$\phi_{a}(x)y=Df(a)y+\cdots,$$
we see $\phi_{a}(x)-\phi_{a}(a)=\phi_{a}(x)-Df(a)=\cdots,$ so when the authors show
$$\lim_{h\rightarrow 0}\frac{\|\epsilon_{a}(h)h^{t}\|}{\|h\|^{2}}=0,$$
they are really proving
$$\lim_{x\rightarrow a}\|\phi_{a}(x)y-\phi_{a}(a)y\|=0,\qquad\forall y\in\mathbb{R}^{n},$$
which is precisely what we want to establish when proving $\phi_{a}(x)$ is continuous at $a$.
To prove that $f$ is continuous at $a$, we want to show that
$$\lim_{x\rightarrow a}f(x)=f(a),$$
where we note that $f$ is a function, not an operator, so we don't need to introduce $y$ as we did previously. From the above we know
$$\lim_{x\rightarrow a}\phi_{a}(x)=\phi_{a}(a)=Df(a),$$
so
$$\lim_{x\rightarrow a} f(x)=\lim_{x\rightarrow a} f(a)+\lim_{x\rightarrow a}\phi_{a}(x)(x-a)=f(a)+Df(a)(a-a)=f(a),$$
as desired.
Thanks GJA ... that was really helpful ...

Still reflecting on what you have said ...

Thanks again ...

Peter
 

FAQ: Help with D&K Lemma 2.2.7 (Hadamard) Proof: Qs on Continuity of \phi_a

1. What is the D&K Lemma 2.2.7 (Hadamard) Proof?

The D&K Lemma 2.2.7 (Hadamard) Proof is a mathematical proof that demonstrates the continuity of a function, specifically the function φa, where a is a constant. This lemma is often used in the study of real analysis and is named after the mathematicians Darboux and Kurzweil.

2. What is the significance of Lemma 2.2.7 in the D&K Lemma 2.2.7 (Hadamard) Proof?

Lemma 2.2.7 is the key component of the D&K Lemma 2.2.7 (Hadamard) Proof. It states that if a function is continuous at a point, then it is also continuous on an interval containing that point. This lemma is essential for proving the continuity of the function φa in the D&K Lemma 2.2.7 (Hadamard) Proof.

3. What is the role of Hadamard in the D&K Lemma 2.2.7 (Hadamard) Proof?

Hadamard is the name given to this lemma because it was originally proven by the French mathematician Jacques Hadamard. His work in real analysis and differential equations laid the foundation for the D&K Lemma 2.2.7 (Hadamard) Proof and its application in various fields of mathematics.

4. What are the main steps in the D&K Lemma 2.2.7 (Hadamard) Proof?

The D&K Lemma 2.2.7 (Hadamard) Proof involves several steps, including proving the continuity of the function φa at a specific point, applying Lemma 2.2.7 to show the continuity of φa on an interval, and using the definition of continuity to establish the continuity of φa on the entire domain. The proof may also involve using properties of limits and the definition of the derivative.

5. How is the D&K Lemma 2.2.7 (Hadamard) Proof applied in mathematics?

The D&K Lemma 2.2.7 (Hadamard) Proof is commonly used in real analysis and differential equations to demonstrate the continuity of functions. It is also applicable in other areas of mathematics, such as functional analysis and topology, where the concept of continuity is crucial for understanding mathematical structures and their properties.

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