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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 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
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
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