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Determine whether a given relation is well defined


Determine whether the function $f : \mathbb{R}^+ \rightarrow \mathbb{Z}$ given 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.


Solution: Recall that every real number which terminates in an infinite string of 0s has a second expansion which terminates in an infinite string of 9s. Any real number which terminates in an infinite string of 0s beginning in the first decimal place will be a counterexample to the functionhood of this relation. In particular, we have $1 = 1.000\ldots$ and $1 = 0.999\ldots$, but $$f(1.000\ldots) = 0 \neq 9 = f(0.999\ldots).$$

Linearity

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