# Help Understanding Andrew Browder's Proposition 8.14

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In summary, the conversation involves a person seeking further help in understanding the proof of Proposition 8.14 in Andrew Browder's book on Mathematical Analysis. They specifically ask for a formal and rigorous demonstration of how Browder's definition of differentiability (Definition 8.9) leads to the equation f(p + h) - f(p) = Lh + r(h) where r(h)/|h| \to 0 as h \to 0. The response mentions that if we define r(h) as f(p + h) - f(p) - Lh, then Definition 8.9 states that the limit of r(h)/h or r(h)/|h| is 0 as h approaches 0
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MHB
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 ...

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

#### Attachments

• Browder - Proposition 8.14 ... .....png
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• Browder - Definition 8.9 ... Differentials ....png
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Peter said:
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 $$\displaystyle \lim_{h\to0}\frac{r(h)}{|h|} = 0$$

Opalg said:
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

## 1. What is Proposition 8.14 in Andrew Browder's work?

Proposition 8.14 is a mathematical statement in Andrew Browder's book "Mathematical Analysis: An Introduction" that deals with the concept of continuity and differentiability of functions.

## 2. What is the significance of Proposition 8.14?

Proposition 8.14 is significant because it provides a necessary and sufficient condition for a function to be continuously differentiable. This is a fundamental concept in mathematical analysis and has many applications in various fields of science and engineering.

## 3. Can you explain the statement of Proposition 8.14 in simpler terms?

Proposition 8.14 states that if a function is continuously differentiable at a point, then it is also differentiable at that point. In other words, if a function is smooth and has no sharp corners or breaks at a point, then it is also differentiable at that point.

## 4. What is the proof of Proposition 8.14?

The proof of Proposition 8.14 involves using the definition of continuity and differentiability, along with the mean value theorem. It can be found in Andrew Browder's book or in many other advanced calculus textbooks.

## 5. How is Proposition 8.14 used in real-world applications?

Proposition 8.14 is used in many areas of science and engineering, such as physics, economics, and optimization problems. It helps in determining the behavior of a function and its derivatives, which is crucial in understanding and solving real-world problems.

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