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I am focused on Chapter 2: Differentiation ... ...

I need help with the proof of Lemma 2.2.3 ... ...

Duistermaat and Kolk's Lemma 2.2.3 and its proof read as follows:View attachment 7796

I do not understand the strategy (or overall idea) of this proof ... how does considering \(\displaystyle a + th \in U\) lead to demonstrating that \(\displaystyle \text{Df}(a)\) is uniquely determined ...

and

how do we get

\(\displaystyle \text{Df}(a) h = \frac{1}{t} ( f ( a + th ) - f(a) ) - \frac{1}{t} \epsilon_a (th)\) ... ... ... ... ... (1)follow from (2.10) ... ... and then, how does (1) lead to ...\(\displaystyle \text{Df}(a) h = \lim_{ t \rightarrow 0 } ( f ( a + th ) - f(a) )\)and, indeed, how does the above show that \(\displaystyle \text{Df}(a)\) is uniquely determined ... .. ?

Hope someone can help ... ...

Peter==========================================================================================The above post mentions (2.10) which is in the notes following Definition 2.2.2 ... so I am providing Definition 2.2.2 and the accompanying notes ... ... as follows ... ...

View attachment 7797

https://www.physicsforums.com/attachments/7798