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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding the proof of Proposition 8.14 ...

Proposition 8.14 reads as follows:

View attachment 9406

In the above proof by Browder we read the following:"... ... Let \(\displaystyle L = \text{df}_p\); then \(\displaystyle f(p + h) - f(p) = Lh + r(h)\) where \(\displaystyle r(h)/|h| \to 0\) as \(\displaystyle h \to 0\) ... .. "

My question is as follows:Can someone please formally and rigorously demonstrate how Browder's definition of differentiability (Definition 8.9) ...

... ... leads to the equation \(\displaystyle f(p + h) - f(p) = Lh + r(h)\) where \(\displaystyle r(h)/|h| \to 0\) as \(\displaystyle h \to 0\) ... ..

Help will be much appreciated ...

Peter

====================================================================NOTE:

The above post mentions Browder's Definition 8.9 ... Definition 8.9 reads as follows:View attachment 9407

Hope that helps ...

Peter

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...

I need some further help in fully understanding the proof of Proposition 8.14 ...

Proposition 8.14 reads as follows:

View attachment 9406

In the above proof by Browder we read the following:"... ... Let \(\displaystyle L = \text{df}_p\); then \(\displaystyle f(p + h) - f(p) = Lh + r(h)\) where \(\displaystyle r(h)/|h| \to 0\) as \(\displaystyle h \to 0\) ... .. "

My question is as follows:Can someone please formally and rigorously demonstrate how Browder's definition of differentiability (Definition 8.9) ...

... ... leads to the equation \(\displaystyle f(p + h) - f(p) = Lh + r(h)\) where \(\displaystyle r(h)/|h| \to 0\) as \(\displaystyle h \to 0\) ... ..

Help will be much appreciated ...

Peter

====================================================================NOTE:

The above post mentions Browder's Definition 8.9 ... Definition 8.9 reads as follows:View attachment 9407

Hope that helps ...

Peter