Multivariable Analysis: Another Question Re: D&K Lemma 2.2.7

[Math Amateur]

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:

Near to the end of the above text D&K write the following:

" ... ... A direct computation gives ##\| \epsilon_a(h) h^t \|_{ Eucl } = \| \epsilon_a(h) \| \| h \|## , hence##\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 ##\phi_a## is continuous at ##a##. ... ... "

My questions are as follows:

Question 1

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

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

How/why does the proof of Hadamard's Lemma 2.2.7 imply that ##f## is continuous at ##a## if ##f## is differentiable at ##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:

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:

Hope the above helps readers understand the context and notation of the post ...

Peter

 

[andrewkirk]

.

" ... ... A direct computation gives##\| \epsilon_a(h) h^t \|_{ Eucl } = \| \epsilon_a(h) \| \| h \|## , (1)

hence

##\lim_{ h \rightarrow 0 } \frac{ \| \epsilon_a(h) h^t \|_{ Eucl } }{ \| h \|^2 } = \lim_{ h \rightarrow 0 } \frac{ \| \epsilon_a(h) \| }{ \| h \| } = 0## (2)This shows that ##\phi_a## is continuous at ##a##. ... ... "

My questions are as follows:

Question 1

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

Let ##h=x-a##. With that substitution we have

$$\frac{ \| \epsilon_a(h) h^t \|_{ Eucl } }{ \| h \|^2 }
= \frac{ \| \epsilon_a(x-a) (x-a)^t \|_{ Eucl } }{ \|x-a \|^2 }
=\|\phi_a(x)-\phi_a(a)\|
$$
per the above definition of ##\phi_a(x)##

So what you have quoted shows that the limit as ##x\to a## of ##\|\phi_a(x)-\phi_a(a)\|## is zero, which is one way of defining continuity of ##\phi_a## at ##a##..

 

[Math Amateur]

Thanks Andrew ...

Appreciate your help ...

Peter