I am trying to understand the concept of limit point in general.
For the open set $(0,1)$, earlier discussions on stackexchange showed that all point in the open set $(0,1)$ is a limit point. Similarly, we can say the same about the closed set $[0,1]$, right ?. For every point $x$ in the set $[0,1]$, we can find $\epsilon$ neighborhood ($V_{\epsilon}$) such that $V_{\epsilon}(x) \cap A \neq \{\emptyset,x\}$.
If the above comment is true, anyone can give an example of a continuous set containing isolated points ?
By "continuous", I think you mean connected. But a connected set might not have all of its limit points.
Here's an example. Consider the set $(0,1)$. It is true that every point in the set is a limit point. For example, let's look at the point $1/2$. Is this a limit point? To be a limit point, we need to create a sequence of points in the set that approach this point. The Sequence $1/2, 1/2, 1/2, ...$ is such a set of points.
What about the point $0$? Is $0$ a limit point of this set? It is! Consider the sequence $1/2, 1/4, 1/8, 1/16, 1/32, ...$ This sequence is getting closer and closer to the point $0$, and no point in this sequence will ever leave the set $(0,1)$. Therefore, the set $(0,1)$ does not contain all its limit points.
Any set that contains all of its limit points is called a "closed" set. So $(0,1)$ is not a compact set. But $[0,1]$ is.