Problem related to continuous complex mapping. 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 $\bar D=D\cup\partial D$. 
1) I want to show that: $\partial(g(D))\subseteq g(\partial D).$
And further, 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.
So basically we have to show that the boundary of the open set $g(D)$ is contained in image of boundary of $D$ (and sometimes strictly contained). I think that we will use open mapping theorem. But how this theorem will help us here that is not clear.

It's better to assume the function $g$ to be bounded. If the image curve $f(\partial D)$ forms infinitely many loops everywhere, then $\partial f(D)\subsetneqq f(\partial D)$.
