if $A\subseteq \mathbb R^n$ is connected then is $A´$ (derived set) connected?

Question

are the following statements true?

1) if $A\subseteq \mathbb R^n$ is connected then is $A´$ connected?

2)if $A´$ is connected then is $A$ connected?

I can´t find any counterexamples. Can you help me please? I would really appreciate it :)

Answer 1

For (1), notice that if $A$ is connected and has at least two points then it cannot have isolated points. So $A' = A \bigcup A' = \overline{A}$ (closure of $A$). It is easy to see that any disconnection of $\overline{A}$ is a disconnection of $A$.

For (2), let $A$ be a any countable dense set.