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- Jun 22, 2012

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I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ...

I need further help with other aspects of Example 1.5, Section 1.2, Chapter III ...

Example 1.5, Section 1.2, Chapter III, reads as follows:

My questions are as follows:

**Question 1**In the above text by Palka we read the following:

" ... ... Recall that the function \(\displaystyle \theta\) is continuous on the set \(\displaystyle D = \mathbb{C} \sim ( - \infty, 0]\) (Lemma II.2.4), a fact that makes it clear that \(\displaystyle f\), too, is continuous in \(\displaystyle D\) ... ... "

How/why exactly does the fact that \(\displaystyle \theta\) is continuous on the set \(\displaystyle D\) imply that \(\displaystyle f\) is continuous in D ... ...

**Question 2**In the above text by Palka we read the following:

" ... ... we observe that \(\displaystyle \lim_{ h \to 0+ } \theta (z_0 - ih) = - \pi\) ... ... "

Can someone please explain how/why \(\displaystyle \lim_{ h \to 0+ } \theta (z_0 - ih) = - \pi\) ... ...

**Question 3**In the above example Palka asserts that \(\displaystyle -i \sqrt{ \mid z_0 \mid } = - \sqrt{z_0}\) ...

Can someone please demonstrate how/why this is the case ...

Help with the above questions will be much appreciated ...

Peter