Dirichlet function
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In mathematics, the Dirichlet function is the indicator function \mathbf{1}_\Q of the set of rational numbers \Q over the set of real numbers \R , i.e. \mathbf{1}_\Q(x) = 1 for a real number if is a rational number and \mathbf{1}_\Q(x) = 0 if is not a rational number (i.e. is an irrational number). \mathbf 1_\Q(x) = \begin{cases} 1 & x \in \Q \\ 0 & x \notin \Q \end{cases} It is named after the mathematician Peter Gustav Lejeune Dirichlet. It is an example of a pathological function which provides counterexamples to many situations.
In mathematics, the Dirichlet function is the indicator function \mathbf{1}_\Q of the set of rational numbers \Q over the set of real numbers \R , i.e. \mathbf{1}_\Q(x) = 1 for a real number if is a rational number and \mathbf{1}_\Q(x) = 0 if is not a rational number (i.e. is an irrational number). \mathbf 1_\Q(x) = \begin{cases} 1 & x \in \Q \\ 0 & x \notin \Q \end{cases} It is named after the mathematician Peter Gustav Lejeune Dirichlet. It is an example of a pathological function which provides counterexamples to many situations. For any real number and any positive rational number , \mathbf{1}_\Q(x + T) = \mathbf{1}_\Q(x) . The Dirichlet function is therefore an example of a real periodic function which is not constant but whose set of periods, the set of rational numbers, is a dense subset of \R . ==See also== Thomae's function, a variation that is discontinuous only at the rational numbers.
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