TITLE: On the existence of a continuous bijection $f\colon [0,1]\to [0,1]\times [0,1]$
QUESTION [6 upvotes]: Let $f$ be a continuous function on $[0,1]$ such that $f([0,1])=[0,1]\times[0,1].$ Then show that $f$ is not one-one.
Hints will be appreciated.

REPLY [4 votes]: Hint: For each $x\in [0,1]^2$, 
$$
[0,1]^2 - \{x\} 
$$ is connected, but
$[0,1] - \{1/2\}$ is not.