I have some problems understanding the following definition
DEFINITION
A set of points A of the xy plane is called connected if any points of A can be joint by a continuous curve lies entirely in A.
A set of points A of the xy plane is called open if each point of A is the center of a circle whose interior lies entirely in A.
An OPEN + CONNECTED set is called a domain.
A point P is called a boundary point of domain D if every circle about P contains both points in D and not in D.
A DOMAIN + BOUNDARY POINTS is called closed domain.
I don't really understand the 2. and 4.
A question for 1.
Take this scenario
If we join $P$ and $Q$ by a straight line there will be a removable discontinuity between them called it $a$. If we connect $P$ and $Q$ with a curvy line such that there is no discontinuity in between then they can be joined. Is this what CONNECTED really means?
For your scenario with (1):
What do you mean by removable discontinuity in this case? The definition 1 simply says that if there exists, for each pair of points in $A$, a curve in $A$ that joins the points, then $A$ is connected. There is no "removable discontinuity" mentioned, and I don't even know how it could enter the discussion here.
For (2):
Which part do you "not understand"? What (2) says is a definition. There is nothing to understand here, you can just accept the definition, and then try some exercises to internalize the definition. By definition, a set $A$ is closed if every point $(x,y)\in A$ is the center of some circle that is in $A$. Let's take an example.
For example, if $A$ is the disc $$A=\{(x,y)| x^2+y^2<1\}$$centered at $0$ and the radius of $1$, then $A$ is open. You can see this by showing that if $(x,y)\in A$, then the circle with the center at $(x,y)$ and the radius of $\min\{1-|x|, 1-|y|\}$ is a subset of $A$.
For $4$, again, I don't know what you mean by "I don't understand", but you can see, for example, that $(1,0)$ is a boundary point of $A$ from above. You can do this by proving the statement: