Solving Kantorovitz' Example 3 on Real Valued Functions of Several Variables

In summary, the conversation is about a specific example in the book "Several Real Variables" by Shmuel Kantorovitz. The example involves the function \phi_x(h) and its value at 0, which is defined by the author to be zero. However, the function is not defined at 0 and the question is raised about how Kantorovitz obtains a value for it in the example. The explanation is provided that we can extend the function by defining f(0) = 1, which creates a continuous and differentiable function. Additional resources on "removable discontinuities" are suggested for further understanding.
  • #1
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I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ...

I am currently focused on Chapter 2: Derivation ... ...

I need help with an aspect of Kantorovitz's Example 3 on pages 65-66 ...

Kantorovitz's Example 3 on pages 65-66 reads as follows:
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In the above example, we read the following:"... ... \(\displaystyle \frac{ \mid \phi_0 (h) \mid }{ \| h \| } = \frac{ \mid h_1 \text{ sin } (h_2 h_3) \mid }{ \| h \|^{ a + 1 } }\) ... ... ... "
My question is as follows:In the Section on The Differential (see scanned text below) ...

Kantorovitz defines \(\displaystyle \phi_x(h)\) as follows:

\(\displaystyle \phi_x(h) := f(x +h) - f(x) - Lh\)

so that

\(\displaystyle \phi_0(h) := f(0 +h) - f(0 ) - Lh = f(h) - f(0)\) ...... BUT in the Example ... as I understand it ... \(\displaystyle f(0)\) does not exist ...? ...

... BUT ... Kantorovitz effectively gives \(\displaystyle \mid \phi_0 (h) \mid = \frac{ \mid h_1 \text{ sin } (h_2 h_3) \mid }{ \| h \| } \)
Can someone please explain how Kantorovitz gets this value when \(\displaystyle f(0)\) does not exist?Help will be much appreciated ...

Peter============================================================================================

***NOTE***

Readers of the above post may be helped by having access to Kantorovitz' Section on "The Differential" ... so I am providing the same ... as follows:
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  • #2
Hi, Peter.

Peter said:
Can someone please explain how Kantorovitz gets this value when \(\displaystyle f(0)\) does not exist?

The function is defined by the author to be zero at zero.

This example is akin to the single-variable function $f(x)=\sin(x)/x$, which, in itself, is not defined at zero. However, by using L'Hopital's rule (or a power series expansion), we can see that the limit of this function as $x$ approaches 0 is 1. Hence, we can extend the original function by defining $f(0)=1.$ Doing so produces a continuous (in fact differentiable) function at $x=0$. Looking up examples of "removable discontinuities" online could prove helpful.
 
  • #3
GJA said:
Hi, Peter.
The function is defined by the author to be zero at zero.

This example is akin to the single-variable function $f(x)=\sin(x)/x$, which, in itself, is not defined at zero. However, by using L'Hopital's rule (or a power series expansion), we can see that the limit of this function as $x$ approaches 0 is 1. Hence, we can extend the original function by defining $f(0)=1.$ Doing so produces a continuous (in fact differentiable) function at $x=0$. Looking up examples of "removable discontinuities" online could prove helpful.
Thanks GJA ...

You write:

" ... ... The function is defined by the author to be zero at zero. ... ... "

Oh! Now how did I miss that ... ! ... :( ...

Thanks for further advice ... most helpful!

Peter
 

FAQ: Solving Kantorovitz' Example 3 on Real Valued Functions of Several Variables

1. What is Kantorovitz' Example 3 on Real Valued Functions of Several Variables?

Kantorovitz' Example 3 is a mathematical problem that involves finding the maximum value of a real valued function of several variables subject to certain constraints.

2. Why is solving Kantorovitz' Example 3 important?

Solving Kantorovitz' Example 3 is important because it helps us understand the concept of optimization in mathematics, which has many practical applications in various fields such as economics, engineering, and physics.

3. What are the steps to solve Kantorovitz' Example 3?

The steps to solve Kantorovitz' Example 3 include setting up the problem by writing out the constraints and the objective function, finding the gradient of the objective function, setting up the Lagrangian equation, and solving for the critical points using the Lagrange multiplier method.

4. What are some common challenges when solving Kantorovitz' Example 3?

Some common challenges when solving Kantorovitz' Example 3 include correctly setting up the constraints and objective function, finding the critical points using the Lagrange multiplier method, and interpreting the results in the context of the problem.

5. How can one improve their skills in solving problems like Kantorovitz' Example 3?

Improving skills in solving problems like Kantorovitz' Example 3 can be achieved by practicing similar problems, seeking guidance from a teacher or mentor, and understanding the underlying concepts and techniques involved in optimization problems.

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