Consider an injective function $\,\,f:[0,1]\rightarrow[0,1].\,$ Then is it true that there always exists some non-empty open subinterval of $[0,1]$, such that $f([0,1])$ is dense in that subinterval? That is to say, do there exist $a$ and $b$ such that $a<b$ and for any $c$, $d$ in $(a,b)$, there exists a $y$ in $(c,d)$ such that $y=f(x)$ for some $x$?
I'd be much obliged if someone could give me ideas on how to go on about proving this or maybe provide a counter-example since I don't even know whether it's true or not.