Does the following function has a maximum?

Question

$f$: [$0$,$1$] $\rightarrow$ $\mathbb{R}$

if x rational: $f$($x$) = $x$

if x irrational: $f$($x$) = $ \frac{1}{x}$

So I think, that this function doesn't have a maximum. If I consider the fact that the $\lim_{x \rightarrow 0}$ $\frac{1}{x}$ = $\infty$ then of course the function does not have a maximum. But how can I show that this function does not a maximum? So I'm searching for a "clean" proof.

Thank you for your reply.

Answer 1

Hint : The set of irrational numbers in the interval $[0,1]$ is dense in $[0,1]$. In other words, there are irrational numbers arbitary close to $0$ and greater than $0$.