Differential for 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:

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 for the function in Example 3 ...? ...

... 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 for $\mid \phi_0 (h) \mid$ 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:

Hope that helps in understanding the post ...

Peter

 

[Math Amateur]

I've just noticed that the author defines the function to be zero at zero!

... careless of me to miss that!

My apologies ...

Peter