1. We are given with a map $g:\bar D\to \Bbb C $, which is continuous on $\bar D$ and analytic on $D$. Where $D$ is a ***bounded domain*** and $\bar D=D\cup\partial D$. Then $\partial(g(D))\subseteq g(\partial D).$(I already know, how to prove it).

I need two examples: 

a) First, to show that the above inclusion can be strict, that is: $\partial(g(D))\not= g(\partial D).$

b) Second example, I need to show that conclusion in (1) is not true if $D$ is not bounded.

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There is an example, I was working on yesterday. But I couldn't understand it completely.

a) If we take $g(z)= z^2$ and $D$ =\begin{cases}z, & \text{where 1<|z|<2} \\\end{cases}
This $g$ is not 1-1. 

Now, *we want to prove that $g(\partial D)\not\subset \partial(g(D)) $.* Therefore, we need to show that $\exists $ some $z\in g(\partial D)$ but $z \not\in \partial(g(D))$. 
How will we show that ???

I want to talk about domain $D$ and its image by map $g$. Please check it:![enter image description here](https://i.stack.imgur.com/DiCYO.png) 

I am not sure that I am taking right domain.
??