Show that $f$ is not constant and find $f$ ?

Question

Given $f:RR->QQ$ continuous in $RR$ with $f(2004)=2013$ . Show that $f$ is not constant and find $f$

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Answer 1 · 2018-03-21T14:14:22

The question should say "Show that $f$ is a constant function."

Use the intermediate value theorem.

**Suppose that ** $f$ is a function with domain $RR$ and $f$ is continuous on $RR$ .

We shall show that the image of $f$ (the range of $f$ ) includes some irrational numbers.

If $f$ is not constant, then there is an $r in RR$ with $f(r) =s != 2013$

But now $f$ is continuous on the closed interval with endpoints $r$ and $2004$ , so $f$ must attain every value between $s$ and $2013$ .

There are irrational numbers between $s$ and $2013$ , so the image of $f$ includes some irrational numbers.