Differentiability of mappings from R to R .... ....

[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 an aspect of the proof of Proposition 2.2.1 ... ...

Duistermaat and Kolk's Proposition 2.2.1 and its proof (including the preceding relevant definition) read as follows:

https://www.physicsforums.com/attachments/7787
https://www.physicsforums.com/attachments/7788

Can someone help me to rigorously prove that $$(ii) \Longrightarrow (iii)$$ ...

Further ... how do we know in doing this that we can, as D&K direct us, take $$L(a) = \phi_a(a)$$ and $$\epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h$$ ... ... Help will be much appreciated ... ...

Peter***EDIT 1***

Reflecting on my own questions I can see regarding my question ... ... how do we know in doing this that we can, as D&K direct us, take $$L(a) = \phi_a(a)$$ ... ... ... that the proposed substitution seems permissible ...

... since in the given equation:

$$f(x) = f(a) + \phi_a (x) (x - a) $$

although \phi_a (x) is a function, $$\phi_a(a) $$ is simply a number $$\in \mathbb{R}$$ ... being the value at the point $$a$$ of a continuous function on and into $$\mathbb{R} $$

... and so presumably we can substitute $$L(a)$$ for $$\phi_a(a) $$ since $$L(a)$$ is also a number ...

Is that right ... ?

Not quite sure what is going on, though ...

Peter

***EDIT 2***

Justification for letting $$\epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h$$ ... ... ... ... basically I think this is justified because $$\epsilon_a(h)$$ is defined as a function on and into $$\mathbb{R}$$ ... and $$( ( \phi_a (a + h) - \phi_a (a) ) h$$ is also such a function ... mind you we would have to show that, given that substitution that we have $$\lim{ h \rightarrow 0 } \frac{ \epsilon_a (h) }{ h } = 0 $$... ...

Is that a correct justification/argument for putting $$\epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h$$ ... ... ?

Peter

 

[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 an aspect of the proof of Proposition 2.2.1 ... ...

Duistermaat and Kolk's Proposition 2.2.1 and its proof (including the preceding relevant definition) read as follows:

Can someone help me to rigorously prove that $$(ii) \Longrightarrow (iii)$$ ...

Further ... how do we know in doing this that we can, as D&K direct us, take $$L(a) = \phi_a(a)$$ and $$\epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h$$ ... ... Help will be much appreciated ... ...

Peter***EDIT 1***

Reflecting on my own questions I can see regarding my question ... ... how do we know in doing this that we can, as D&K direct us, take $$L(a) = \phi_a(a)$$ ... ... ... that the proposed substitution seems permissible ...

... since in the given equation:

$$f(x) = f(a) + \phi_a (x) (x - a) $$

although \phi_a (x) is a function, $$\phi_a(a) $$ is simply a number $$\in \mathbb{R}$$ ... being the value at the point $$a$$ of a continuous function on and into $$\mathbb{R} $$

... and so presumably we can substitute $$L(a)$$ for $$\phi_a(a) $$ since $$L(a)$$ is also a number ...

Is that right ... ?

Not quite sure what is going on, though ...

Peter

***EDIT 2***

Justification for letting $$\epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h$$ ... ... ... ... basically I think this is justified because $$\epsilon_a(h)$$ is defined as a function on and into $$\mathbb{R}$$ ... and $$( ( \phi_a (a + h) - \phi_a (a) ) h$$ is also such a function ... mind you we would have to show that, given that substitution that we have $$\lim{ h \rightarrow 0 } \frac{ \epsilon_a (h) }{ h } = 0 $$... ...

Is that a correct justification/argument for putting $$\epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h$$ ... ... ?

Peter

After reflecting on my post I have formulated a proof of $$(ii) \Longrightarrow (iii)$$ ... ... but am unsure of the validity of my proof ... I would be grateful if someone could critique my proof and either confirm its correctness and/or point out errors and shortcomings ...
We are given the following equation:

$$f(x) = f(a) + \phi_a (x) (x - a) $$ ... ... ... ... ... (1)Now $$(1) \Longrightarrow \phi_a(x) = \frac{ f(x) - f(a) }{ x - a }$$now ... put $$x = a + h$$ ... ...Then $$\phi_a( a + h ) = \frac{ f( a + h ) - f(a) }{ h } $$But we have $$\phi_a(a) = f'(a)$$ ... ... by definition ...Thus from the above analysis we have ... ...$$\epsilon_h (h) = ( \phi_a( a + h) - \phi_a (a) ) h$$ $$\Longrightarrow \epsilon_h (h) = \left( \frac{ f( a + h ) - f(a) }{ h } - f'(a) \right) h $$
Thus ... $$ \lim_{h \rightarrow 0 } \frac{ \epsilon_a (h) }{h} \ = \ \lim_{h \rightarrow 0 } \left( \frac{ f( a + h ) - f(a) }{ h } - f'(a) \right)$$and so $$\lim_{h \rightarrow 0 } \frac{ \epsilon_a (h) }{h} = f'(a) - f'(a) = 0$$ as required ... ...
Can someone please confirm that the above proof is correct and/or point out any errors or shortcomings ...

Peter