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

[Math Amateur]

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:"... ... $$\frac{ \mid \phi_0 (h) \mid }{ \| h \| } = \frac{ \mid h_1 \text{ sin } (h_2 h_3) \mid }{ \| h \|^{ a + 1 } }$$ ... ... ... "
My question is as follows:In the Section on The Differential (see scanned text below) ...

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

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

so that

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

... BUT ... Kantorovitz effectively gives $$\mid \phi_0 (h) \mid = \frac{ \mid h_1 \text{ sin } (h_2 h_3) \mid }{ \| h \| } $$
Can someone please explain how Kantorovitz gets this value when $$f(0)$$ 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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[GJA]

Hi, Peter.

Can someone please explain how Kantorovitz gets this value when $$f(0)$$ 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.

 

[Math Amateur]

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