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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 $x \in (\mathbb{R} - \mathbb{Q})$.

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Can you write down the definition of Dirichlet's function or add a link to a description elsewhere (Wikipedia/MathWorld...)? – Srivatsan Jan 14 '12 at 10:15
Yes, here it is link – andreas.vitikan Jan 14 '12 at 10:20
I don't know what is meant by the accumulation points of a function. – Gerry Myerson Jan 14 '12 at 10:55
Accumulation point, apparently in English they're also called limit points. – andreas.vitikan Jan 14 '12 at 10:58
The function is discontinuous everywhere, so its set of discontinuity points is the set of all reals, so the accumulation points of the set of discontinuity points is the accumulation points of the reals, which is the reals. But we really need some clarification to the question. What exactly is it that you want explained? Property 6? Property 7? how 7 is deduced from 6? We want to help you, but we can't read your mind to work out what you really want to know. – Gerry Myerson Jan 14 '12 at 23:30

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