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Recall that the set $\mathbb{Q}$ of rational numbers is countable; thus we can put them all in a sequence ({ i.e.}~we can enumerate them). Let $(r_n)$ be such a sequence and let $f(x)=\begin{cases} \frac{1}{n} &\text{ if } x=r_n\in \mathbb{Q}\\ 0 &\text{ if }x\notin \mathbb{Q}\end{cases}$.\

(1) Is this function injective (one-to-one)? Why or why not?

(2) Is $f(\mathbb{R})$ (the image/range of $f$) compact? Justify your answer.

(3) Which of the following preimages are countable: $f^{-1}(\{1/2\})$, $f^{-1}([0,1/2])$, $f^{-1}([1/2,1])$? Justify your answers.

(4) Prove that $f$ is continuous at every $x\in \mathbb{R}-\mathbb{Q}$ and discontinuous at every $x\in \mathbb{Q}$.

Response: I haven't got any work to show for any of them.

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    Not even for (1)? Do you think that there might be more than one irratonal number? For (3), can you at least describe the first of the three preimages?2012-11-13

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