Interactive Real Analysis

• Real Analysis
• 1. Sets and Relations
• 1.1. Notation and Set Theory
• 1.2. Relations and Functions
• 1.3. Equivalence Relations and Classes
• 1.4. Natural Numbers, Integers, and Rational Numbers
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

1.2. Relations and Functions

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 Dirichlet's 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)

Back

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}.