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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:

... 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 \(\displaystyle a\). ... ...?

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

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: