Exercise 0.1.6 (Well defined function)

Question

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

Richard Ganaye · Accepted Answer

\begin{proof} 
Since some real have two decimal representations, $f$ is not well defined.

For instance $0.999999999\ldots$ has image $9$ and $1 = 1.0000000\ldots$ has image $0$, but $1 = 0.99999999\ldots$.

To obtain a well defined function, we must reject all decimal representations which end with an infinity of consecutive $9$ as improper representations.
\end{proof}

Exercise 0.1.6 (Well defined function)

Answers

Comments

Exercise 0.1.6 (Well defined function)

Answers

Comments

Add answer