Interactive Real Analysis - part of MathCS.org

Next | Previous | Glossary | Map | Discussion

Example 1.2.5(a):

Let f(x) = 0 if x is rational and f(x) = 1 if x is irrational. This function is called Dirichletís Function. The range for f is R. Find the image of the domain of the Dirichlet Function when:
  1. the domain of f is Q
  2. the domain of f is R
  3. the domain of f is [0, 1] (the closed interval between 0 and 1)
  1. When the domain is Q, we have that f(x) = 0 for any x, because x must be a rational number. Hence, the image of the domain Q is the set consisting of the single element {0}.

  2. When the domain is R, we have that f(x) could be 0 or 1, because x could be rational or irrational. f(x) can not be any other number. Hence, the image of the domain R is the set consisting of the two elements {0, 1}.

  3. The interval [0, 1] contains irrationals as well as rational numbers. Therefore, f(x) could be equal to 0 or 1, and the image of [0, 1] under f is the set with the two elements {0, 1}.
Next | Previous | Glossary | Map | Discussion