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**Differentialbility & Continuity of Multivariable Vector-Valued Functions ... D&K Lemma 2.2.7 ...**

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 Lemma 2.2.7 (Hadamard...) ... ...

Duistermaat and Kolk's Lemma 2.2.7 and its proof read as follows:https://www.physicsforums.com/attachments/7829

View attachment 7830In the above proof we read the following:

" ... ... Or, in other words since \(\displaystyle (x - a)^t y = \langle x - a , y \rangle \in \mathbb{R}\) for \(\displaystyle y \in \mathbb{R}^n\),\(\displaystyle \phi_a(x) y = Df(a)y + \frac{ \langle x - a , y \rangle }{ \| x - a \|^2 } \epsilon_a ( x _ a ) \in \mathbb{R}^p \ \ \ \ \ (x \in U \setminus \{ a \} , \ y \in \mathbb{R}^n )\).

Now indeed we have \(\displaystyle f(x) = f(a) + \phi_a(x) ( x - a )\). ... ... "

My question is as followsow/why does \(\displaystyle \phi_a(x) y = Df(a)y + \frac{ \langle x - a , y \rangle }{ \| x - a \|^2 } \epsilon_a ( x _ a ) \in \mathbb{R}^p\)

... imply that ...

\(\displaystyle f(x) = f(a) + \phi_a(x) ( x - a )\) ... ... ... ?

Help will be much appreciated ...

Peter

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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 7831

View attachment 7832

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 7833

View attachment 7834

View attachment 7835Hope the above helps readers understand the context and notation of the post ...

Peter

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