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- Jun 22, 2012

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I need some help with an aspect of the proof of Theorem 7.1 ...

The statement of Theorem 7.1 reads as follows:

At the start of the above proof by Markushevich we read the following:

"If \(\displaystyle f(z)\) has a derivative \(\displaystyle f'_E(z_0)\) at \(\displaystyle z_0\), then by definition

\(\displaystyle \frac{ \Delta_E f(z) }{ \Delta z } = f'_E(z_0) + \epsilon ( z, z_0 )\)

where \(\displaystyle \epsilon ( z, z_0 ) \to 0\) as \(\displaystyle \Delta z \to 0\). ... ... "

Now previously in Equation 7.1 at the start of Chapter 7, Markushevich has defined \(\displaystyle f'_E(z_0)\) as follows:

\(\displaystyle f'_E(z_0) = \frac{ f(z) - f(z_0) }{ z - z_0 } = \frac{ \Delta_E f(z) }{ \Delta z }\) ... ... ... (1)

How exactly (formally and rigorously) is equation (1) exactly the same as \(\displaystyle \frac{ \Delta_E f(z) }{ \Delta z } = f'_E(z_0) + \epsilon ( z, z_0 )\) ...

... strictly speaking, shouldn't Markushevich be deriving \(\displaystyle \frac{ \Delta_E f(z) }{ \Delta z } = f'_E(z_0) + \epsilon ( z, z_0 )\) ... from equation (1) ...

Peter