mapping

[logistic_guy]

Determine whether the function \(\displaystyle f : \mathbb{R}^{+} \rightarrow \mathbb{Z}\) defined by mapping a real number \(\displaystyle r\) to the first digit to the right of the decimal point in a decimal expansion of \(\displaystyle r\) is well defined.

 

[logistic_guy]

The idea of this exercise is this:

\(\displaystyle f(r_1) = f(0.123\cdots) = 1\)
\(\displaystyle f(r_2) = f(5.419\cdots) = 4\)
\(\displaystyle f(r_3) = f(2.871\cdots) = 8\)

Do you think that this weird function is well defined? Or there's an example that breaks it down?

Think about itfor a little before I hand you over the answer in the next post!

 

[logistic_guy]

We know that \(\displaystyle 0.999\cdots = 1.000\cdots\)

\(\displaystyle f(0.999\cdots) = 9\)
\(\displaystyle f(1.000\cdots) = 0\)

Since \(\displaystyle f(0.999\cdots) \neq f(1.000\cdots)\), the function \(\displaystyle f\) is not well defined.