Differentiability of mappings from R to R .... ....

In summary: In order to do this, we need to show that \lim_{h\to 0} \epsilon_a(h)/h = 0. From the definition of \epsilon_a(h), we can see that it is a function on and into \mathbb{R}. So, in order to show that the limit is 0, we need to show that \epsilon_a(h) approaches 0 as h approaches 0. And since \epsilon_a(h) is a function, we need to show that the value of the function at h = 0 is 0. This is where we can substitute L(a) for \phi_a(a).Since L(a) is a linear map, we know that L(a
  • #1
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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 2.2.1 ... ...

Duistermaat and Kolk's Proposition 2.2.1 and its proof (including the preceding relevant definition) read as follows:

https://www.physicsforums.com/attachments/7787
https://www.physicsforums.com/attachments/7788

Can someone help me to rigorously prove that \(\displaystyle (ii) \Longrightarrow (iii)\) ...

Further ... how do we know in doing this that we can, as D&K direct us, take \(\displaystyle L(a) = \phi_a(a)\) and \(\displaystyle \epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h\) ... ... Help will be much appreciated ... ...

Peter***EDIT 1***

Reflecting on my own questions I can see regarding my question ... ... how do we know in doing this that we can, as D&K direct us, take \(\displaystyle L(a) = \phi_a(a)\) ... ... ... that the proposed substitution seems permissible ...

... since in the given equation:

\(\displaystyle f(x) = f(a) + \phi_a (x) (x - a) \)

although \phi_a (x) is a function, \(\displaystyle \phi_a(a) \) is simply a number \(\displaystyle \in \mathbb{R}\) ... being the value at the point \(\displaystyle a\) of a continuous function on and into \(\displaystyle \mathbb{R} \)

... and so presumably we can substitute \(\displaystyle L(a)\) for \(\displaystyle \phi_a(a) \) since \(\displaystyle L(a)\) is also a number ...

Is that right ... ?

Not quite sure what is going on, though ...

Peter

***EDIT 2***

Justification for letting \(\displaystyle \epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h\) ... ... ... ... basically I think this is justified because \(\displaystyle \epsilon_a(h)\) is defined as a function on and into \(\displaystyle \mathbb{R}\) ... and \(\displaystyle ( ( \phi_a (a + h) - \phi_a (a) ) h\) is also such a function ... mind you we would have to show that, given that substitution that we have \(\displaystyle \lim{ h \rightarrow 0 } \frac{ \epsilon_a (h) }{ h } = 0 \)... ...

Is that a correct justification/argument for putting \(\displaystyle \epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h\) ... ... ?

Peter
 
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  • #2
Peter said:
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 2.2.1 ... ...

Duistermaat and Kolk's Proposition 2.2.1 and its proof (including the preceding relevant definition) read as follows:

Can someone help me to rigorously prove that \(\displaystyle (ii) \Longrightarrow (iii)\) ...

Further ... how do we know in doing this that we can, as D&K direct us, take \(\displaystyle L(a) = \phi_a(a)\) and \(\displaystyle \epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h\) ... ... Help will be much appreciated ... ...

Peter***EDIT 1***

Reflecting on my own questions I can see regarding my question ... ... how do we know in doing this that we can, as D&K direct us, take \(\displaystyle L(a) = \phi_a(a)\) ... ... ... that the proposed substitution seems permissible ...

... since in the given equation:

\(\displaystyle f(x) = f(a) + \phi_a (x) (x - a) \)

although \phi_a (x) is a function, \(\displaystyle \phi_a(a) \) is simply a number \(\displaystyle \in \mathbb{R}\) ... being the value at the point \(\displaystyle a\) of a continuous function on and into \(\displaystyle \mathbb{R} \)

... and so presumably we can substitute \(\displaystyle L(a)\) for \(\displaystyle \phi_a(a) \) since \(\displaystyle L(a)\) is also a number ...

Is that right ... ?

Not quite sure what is going on, though ...

Peter

***EDIT 2***

Justification for letting \(\displaystyle \epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h\) ... ... ... ... basically I think this is justified because \(\displaystyle \epsilon_a(h)\) is defined as a function on and into \(\displaystyle \mathbb{R}\) ... and \(\displaystyle ( ( \phi_a (a + h) - \phi_a (a) ) h\) is also such a function ... mind you we would have to show that, given that substitution that we have \(\displaystyle \lim{ h \rightarrow 0 } \frac{ \epsilon_a (h) }{ h } = 0 \)... ...

Is that a correct justification/argument for putting \(\displaystyle \epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h\) ... ... ?

Peter
After reflecting on my post I have formulated a proof of \(\displaystyle (ii) \Longrightarrow (iii)\) ... ... but am unsure of the validity of my proof ... I would be grateful if someone could critique my proof and either confirm its correctness and/or point out errors and shortcomings ...
We are given the following equation:

\(\displaystyle f(x) = f(a) + \phi_a (x) (x - a) \) ... ... ... ... ... (1)Now \(\displaystyle (1) \Longrightarrow \phi_a(x) = \frac{ f(x) - f(a) }{ x - a }\)now ... put \(\displaystyle x = a + h\) ... ...Then \(\displaystyle \phi_a( a + h ) = \frac{ f( a + h ) - f(a) }{ h } \)But we have \(\displaystyle \phi_a(a) = f'(a)\) ... ... by definition ...Thus from the above analysis we have ... ...\(\displaystyle \epsilon_h (h) = ( \phi_a( a + h) - \phi_a (a) ) h\) \(\displaystyle \Longrightarrow \epsilon_h (h) = \left( \frac{ f( a + h ) - f(a) }{ h } - f'(a) \right) h \)
Thus ... \(\displaystyle \lim_{h \rightarrow 0 } \frac{ \epsilon_a (h) }{h} \ = \ \lim_{h \rightarrow 0 } \left( \frac{ f( a + h ) - f(a) }{ h } - f'(a) \right)\)and so \(\displaystyle \lim_{h \rightarrow 0 } \frac{ \epsilon_a (h) }{h} = f'(a) - f'(a) = 0\) as required ... ...
Can someone please confirm that the above proof is correct and/or point out any errors or shortcomings ...

Peter
 
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  • #3
Hello Peter,

I can definitely help you with your questions regarding Proposition 2.2.1 in Duistermaat and Kolk's book. Let's start with your first question: how do we know that we can substitute L(a) for \phi_a(a)?

First, let's take a look at the definition of L(a) and \phi_a(a). L(a) is defined as the linear map from \mathbb{R}^n to \mathbb{R} given by L(a)(h) = f(a) + \phi_a(h)h. On the other hand, \phi_a(a) is defined as the value of the function \phi_a at the point a. So, while L(a) is a linear map, \phi_a(a) is just a number.

Now, in the proof of Proposition 2.2.1, we are trying to show that (ii) \Longrightarrow (iii). In order to do so, we need to show that L(a) = \phi_a(a). This means that we need to show that the linear map L(a) and the value \phi_a(a) are equal. But since \phi_a(a) is just a number, it is also a constant function. And since L(a) is a linear map, it can also be seen as a constant function. So, in this case, L(a) and \phi_a(a) are the same type of object, making it possible to substitute one for the other.

Moving on to your second question: how do we know that we can let \epsilon_a(h) = ( \phi_a (a + h) - \phi_a (a) ) h?

As you correctly pointed out, \epsilon_a(h) is defined as a function on and into \mathbb{R}. And ( ( \phi_a (a + h) - \phi_a (a) ) h is also a function on and into \mathbb{R}. In fact, it is the function that measures the difference between the values of \phi_a at points a and a+h, multiplied by the difference in the points (h). This is exactly what we want \epsilon_a(h) to be, as it is a measure of the error between L(a) and \phi_a(a). So, it is justified to let \epsilon_a(h) = ( \phi_a (a + h) -
 

Related to Differentiability of mappings from R to R .... ....

What is differentiability of mappings from R to R?

Differentiability of mappings from R to R refers to the property of a function that allows it to have a well-defined derivative at every point in its domain. In simpler terms, it means that the function is smooth and has no sharp or abrupt changes in its graph.

Why is differentiability important in mathematics and science?

Differentiability is important because it allows us to calculate the slope or rate of change of a function at any given point. This is useful in many applications, such as physics, economics, and engineering, where we need to understand how a variable is changing over time or distance.

What is the difference between differentiability and continuity?

Continuity and differentiability are related concepts, but they are not the same. A function is continuous if it has no breaks or gaps in its graph, while a function is differentiable if it has a well-defined derivative at every point. A function can be continuous without being differentiable, but if a function is differentiable, it must also be continuous.

What are the conditions for a function to be differentiable?

In order for a function to be differentiable, it must be continuous at every point in its domain and have a well-defined derivative at every point. This means that the function must have no breaks or gaps in its graph and must not have any sharp or abrupt changes. Additionally, the function must be differentiable from both the left and right at every point.

Can a function be differentiable at a point but not on its entire domain?

Yes, it is possible for a function to be differentiable at a point but not on its entire domain. This can happen if the function has a sharp or abrupt change at a certain point, making it not differentiable at that point. However, the function can still be differentiable at other points in its domain.

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