Rationally continuous

Calculus Level 3

A continuous function $\phi\colon\Bbb R\to\Bbb R$ takes only rational values.

What can be said about $\phi$ ?

Explicitly speaking, what can be said about $\operatorname{Im}\phi$ ?

Details and Assumptions:

$\operatorname{Im}\phi$ denotes the range of $\phi$ , i.e., the image of $\Bbb R$ under the map $\phi$ .
$\Bbb R$ is considered with the usual (metric) topology.

It is a singleton set. It is an open subset of

\Bbb R

It is empty. It has only two elements. It is a countable set. It is a dense subset of

\Bbb R

Trick question: Such a function doesn't exist.

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