I was studying some trigonometry and real analysis and real books, and in the part where periodic functions were discussed, they mentioned Dirichlet's function. Among the properties given for this function was that it discontinuous (and therefore underivable) in any point. This was property 6. Property 7, claimed that it was deduced from the previous property and it said that \mathbb{Q}' = (\mathbb{R} - \mathbb{Q})' = \mathbb{R}' = \bar{\mathbb{R}}. Or in other words, for this function $f: \mathbb{R} \rightarrow \mathbb{R}$ the accumulation points is closed real line $\bar{\mathbb{R}} = \mathbb{R} \cup \{-\infty, \infty\}$. This is correct, and sounds reasonable to me, but can someone provide an explanation? It doesn't have to be very tehnical (I'm not very advanced yet).
Dirichlet's function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined such that $f(x) = 1$ if $x \in \mathbb{Q}$ and $f(x) = 0$ if $x \in (\mathbb{R} - \mathbb{Q})$.