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**Existence of Partial Derivatives and Continuity ... Kantorovitz's Proposition pages 61-62 ...**

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ...

I am currently focused on Chapter 2: Derivation ... ...

I need help with another element of the proof of Kantorovitz's Proposition on pages 61-62 ...

Kantorovitz's Proposition on pages 61-62 reads as follows:View attachment 7808

https://www.physicsforums.com/attachments/7809In the above proof we read the following:

" ... ... Formula 2.4 is trivially true in case \(\displaystyle h_j = 0\), and by (2.2) - (2.4)

\(\displaystyle f(x + h) - f(x) = \sum_{ j = 1}^k [ F_j ( h_j ) - F_j (0) ]\)

\(\displaystyle = \sum_j h_j \frac{ \partial f }{ \partial x_j } ( x + h^{ j - 1 } + \theta_j h_j e^j )\) ... ... ... ... ... "I have tried to derive \(\displaystyle f(x + h) - f(x) = \sum_{ j = 1}^k [ F_j ( h_j ) - F_j (0) ]\) but did not succeed ...

... can someone please show how \(\displaystyle f(x + h) - f(x)\) equals \(\displaystyle \sum_{ j = 1}^k [ F_j ( h_j ) - F_j (0) ]\) ...Also can someone show how the above equals \(\displaystyle \sum_j h_j \frac{ \partial f }{ \partial x_j } ( x + h^{ j - 1 } + \theta_j h_j e^j )\) ... ...

Help will be much appreciated ... ...

Peter

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