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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 6566 ...
Kantorovitz's Example 3 on pages 6566 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
I am currently focused on Chapter 2: Derivation ... ...
I need help with an aspect of Kantorovitz's Example 3 on pages 6566 ...
Kantorovitz's Example 3 on pages 6566 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
Attachments

Kantorovitz  1  Example 3 ... Page 65 ... PART 1 ... .png15.3 KB · Views: 792

Kantorovitz  2  Example 3 ... Page 65 ... PART 2 ... .png17.1 KB · Views: 368

Kantorovitz  1  Sectiion on the DIfferential ... PART 1 ... .png27.4 KB · Views: 381

Kantorovitz  2  Sectiion on the DIfferential ... PART 2 ... .png34.9 KB · Views: 377

Kantorovitz  3  Sectiion on the DIfferential ... PART 3 ... .png13.2 KB · Views: 302