# Show that f is not constant and find f?

## Given $f : \mathbb{R} \to \mathbb{Q}$ continuous in $\mathbb{R}$ with $f \left(2004\right) = 2013$. Show that $f$ is not constant and find $f$

Mar 21, 2018

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

#### Explanation:

Use the intermediate value theorem.

Suppose that $f$ is a function with domain $\mathbb{R}$ and $f$ is continuous on $\mathbb{R}$.

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 $r \in \mathbb{R}$ with $f \left(r\right) = s \ne 2013$

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.