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I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...

I am focused on Chapter 16: Cauchy's Theorem and the Residue Calculus ...

I need help in order to fully understand a remark of Apostol in Section 16.1 ...

The particular remark reads as follows:

View attachment 9279Could someone please demonstrate (in some detail) how it is the case that the complex function \(\displaystyle f\) has a derivative at \(\displaystyle 0\) but at no other point of \(\displaystyle \mathbb{C}\) ... ...Help will be much appreciated ...

Peter

I am focused on Chapter 16: Cauchy's Theorem and the Residue Calculus ...

I need help in order to fully understand a remark of Apostol in Section 16.1 ...

The particular remark reads as follows:

View attachment 9279Could someone please demonstrate (in some detail) how it is the case that the complex function \(\displaystyle f\) has a derivative at \(\displaystyle 0\) but at no other point of \(\displaystyle \mathbb{C}\) ... ...Help will be much appreciated ...

Peter