# Differentiabilty & Maxima of Vector-Valued Functions ... Browder, Proposition 8.14 ... ...

#### Peter

##### Well-known member
MHB Site Helper
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:

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

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NOTE:

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

Hope that helps ...

Peter

#### Opalg

##### MHB Oldtimer
Staff member
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 $$\displaystyle h \to 0$$ ... ..
If you define $r(h)$ by $r(h) = f(p + h) - f(p) - Lh$ then Definition 8.9 says that $$\displaystyle \lim_{h\to0}\frac{r(h)}h = 0$$, or equivalently $$\displaystyle \lim_{h\to0}\frac{r(h)}{|h|} = 0$$

#### Peter

##### Well-known member
MHB Site Helper
If you define $r(h)$ by $r(h) = f(p + h) - f(p) - Lh$ then Definition 8.9 says that $$\displaystyle \lim_{h\to0}\frac{r(h)}h = 0$$, or equivalently $$\displaystyle \lim_{h\to0}\frac{r(h)}{|h|} = 0$$

Thanks Opalg ..

I appreciate the help ...

Peter