Q:

How to define a continuous map from $[0,1] $ onto/into $\mathbb R$?

It is possible to define a continuous map from $[0,1]$ onto $\mathbb R$?

A:

You can't define such a continuous map.
Hint : 
$\bullet$  $[0,1]$ is compact but $\mathbb R$ is not.
$\bullet$  If $f$ is onto then $f([0,1])=\mathbb R$.

