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

[Math Amateur]

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

 

[Jhoneine]

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.

 

[math771]

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!