Multivariable Analysis .... Directional and Partial Derivatives .... D&K Propostion 2.3.3 ....

[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 2.3.2 ... ...

Duistermaat and Kolk's Proposition 2.3.2 and its proof read as follows:
View attachment 7847
https://www.physicsforums.com/attachments/7848
In the above proof by D&K we read the following:

" ... ... Assertion (i) follows from Formula (2.11). ... ..."Can someone please demonstrate (formally and rigorously) that this is the case ... that is that assertion (i) follows from Formula (2.11). ... ...Help will be appreciated ...

Peter==========================================================================================***NOTE***

It may help readers of the above post to have access to the start of Section "2.3: Directional and Partial Derivatives" ... in order to understand the context and notation of the post ... so I am providing the same ... as follows:https://www.physicsforums.com/attachments/7849Hope that the above helps readers of the post understand the context and notation of the post ...

Peter

 

[GJA]

Hi, Peter.

" ... ... Assertion (i) follows from Formula (2.11). ... ..."

Can someone please demonstrate (formally and rigorously) that this is the case ... that is that assertion (i) follows from Formula (2.11). ... ...

Try dividing the first line of (2.11) through by $t$. Now take the limit as $t\rightarrow 0$, use the definition of directional derivative and the second part of (2.11) to obtain the desired result.

 

[math771]

Hello Peter,

I am not familiar with the book you are reading, but I can try to provide some assistance with your question.

Firstly, let's define the notation used in the proof. In Formula (2.11), we have the expression $\partial_i f(x)$, which represents the $i$th partial derivative of $f$ at the point $x$. This can also be written as $\frac{\partial f}{\partial x_i}(x)$.

Now, let's look at Assertion (i) in the proof. It states that for any fixed $i$, the function $\partial_i f$ is continuous. In other words, for any point $x_0$, the limit $\lim_{x \to x_0} \partial_i f(x)$ exists. This can be proved using Formula (2.11) as follows:

By definition, the limit $\lim_{x \to x_0} \partial_i f(x)$ exists if and only if for any given $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $|x - x_0| < \delta$, we have $|\partial_i f(x) - \partial_i f(x_0)| < \epsilon$.

Now, using Formula (2.11), we have $\partial_i f(x) - \partial_i f(x_0) = \frac{\partial f}{\partial x_i}(x) - \frac{\partial f}{\partial x_i}(x_0)$. By the mean value theorem, there exists a point $c$ between $x$ and $x_0$ such that $\frac{\partial f}{\partial x_i}(x) - \frac{\partial f}{\partial x_i}(x_0) = \frac{\partial^2 f}{\partial x_i^2}(c)(x-x_0)$. Since $f$ is continuously differentiable, $\frac{\partial^2 f}{\partial x_i^2}$ is continuous. Hence, for any given $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $|x-x_0| < \delta$, we have $|\frac{\partial^2 f}{\partial x_i^2}(c)| < \epsilon$. Therefore, $|\partial_i f(x) - \partial_i f(x_0)| < \epsilon$,