Interactive Real Analysis
- part of Example 1.2.5(b):
Let f(x) = 0 if x is rational and f(x) = 1 if
x is irrational. This function is called Dirichlet’s Function.
The domain and range for f is R.
Back
The preimage of R is the set of all elements such that f(x)
is contained in R. Since R includes the numbers 0 and 1, the
preimage of R under f is everything in the domain, i.e. R.
- What is the preimage of R ? What is the preimage of [-1/2, 1/2] ?
BackThe preimage of [-1/2, 1/2] consists of all elements such that f(x) is contained in [-1/2, 1/2]. This set does not contain 1, so the preimage of that set can not contain any irrational number. Hence, the preimage of [-1/2, 1/2] is the set of rational numbers Q.