# Proposition 2.2.9: Understanding the Implication of Lemma 1.1.7 (iv) for Peter

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In summary, Proposition 2.2.9 is a statement being considered and Lemma 1.1.7 (iv) is a previously established mathematical statement referenced in Proposition 2.2.9. Peter is involved in the context of Proposition 2.2.9 and Lemma 1.1.7 (iv) has implications for his understanding of it. The implication of Lemma 1.1.7 (iv) for Peter is that he can make conclusions or inferences about Proposition 2.2.9. Proposition 2.2.9 and Lemma 1.1.7 (iv) are connected because Lemma 1.1.7 (iv) is used to support or prove the validity of Proposition
Math Amateur
Gold Member
MHB
I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...

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

I need help with an aspect of the proof of Proposition 2.2.9 ... ...

Duistermaat and Kolk's Proposition 2.2.9 and its proof read as follows:
View attachment 7844
In the above text D&K state that Lemma 1.1.7 (iv) implies Proposition 2.2.9 ...

Can someone please indicate how/why ths is the case ...

Peter
===========================================================================================The above post mentions Lemma 1.1.7 ... so I am providing the text of the same ... as follows:
View attachment 7845
View attachment 7846

Lemma 1.1.7 (iv): Let D be an open set in R^n, let f: D -> R^m be a continuously differentiable function and let x be an interior point of D. Suppose that the partial derivatives of f exist at x and are continuous in a neighbourhood of x. Then there is a neighbourhood U of x such that f is a local diffeomorphism on U; i.e., for all y in U, the derivative of f at y has rank m.===========================================================================================To answer Peter's question, we can see from Lemma 1.1.7 (iv) that if the partial derivatives of f exist at x and are continuous in a neighbourhood of x, then there is a neighbourhood U of x such that f is a local diffeomorphism on U. This is exactly what D&K state in Proposition 2.2.9, which is that if the partial derivatives of f exist at x and are continuous in a neighbourhood of x, then f is a local diffeomorphism. Therefore, Lemma 1.1.7 (iv) implies Proposition 2.2.9.

Lemma 1.1.7: Let E be a measurable set in R^n and let f be a measurable function on E. Then for almost every x in E, the limit

lim (h->0) [f(x+h) - f(x)] / h

exists in [-∞, ∞].

(iv) If f is continuous at x, then the above limit equals f'(x).

Hi Peter,

I can see how Lemma 1.1.7 (iv) would apply to Proposition 2.2.9. In Proposition 2.2.9, D&K are discussing the differentiability of a function f at a point x. They are trying to show that the limit

lim (h->0) [f(x+h) - f(x)] / h

exists, which would imply that f is differentiable at x. Now, in Lemma 1.1.7 (iv), it is stated that if f is continuous at x, then the above limit equals f'(x). This means that if f is continuous at x, then f is differentiable at x, since the limit exists. Therefore, Lemma 1.1.7 (iv) implies Proposition 2.2.9.

I hope this helps clarify the connection between the two statements. Let me know if you need any further assistance. Good luck with your studies!

## 1. What is Proposition 2.2.9 and Lemma 1.1.7 (iv)?

Proposition 2.2.9 is a statement or claim that is being put forward for consideration. Lemma 1.1.7 (iv) is a previously established mathematical statement that is being referenced in Proposition 2.2.9.

## 2. Who is Peter and why is Lemma 1.1.7 (iv) important for him?

Peter is a person or entity that is involved in the context of Proposition 2.2.9. Lemma 1.1.7 (iv) is important for Peter because it has implications or consequences for his understanding of Proposition 2.2.9.

## 3. Can you explain the implication of Lemma 1.1.7 (iv) for Peter in simpler terms?

The implication of Lemma 1.1.7 (iv) for Peter means that based on the previously established mathematical statement, Peter can make conclusions or inferences about Proposition 2.2.9.

## 4. How does Proposition 2.2.9 and Lemma 1.1.7 (iv) relate to each other?

Proposition 2.2.9 and Lemma 1.1.7 (iv) are connected because Lemma 1.1.7 (iv) is being used to support or prove the validity of Proposition 2.2.9.

## 5. What is the significance of understanding the implication of Lemma 1.1.7 (iv) for Peter?

Understanding the implication of Lemma 1.1.7 (iv) for Peter allows him to have a deeper understanding and insight into the validity and importance of Proposition 2.2.9. It also allows him to make informed decisions or conclusions based on the supporting evidence provided by Lemma 1.1.7 (iv).

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