There exists a continuous function f:[0,1] onto [0,10], but there exists no continuous function g:[0,1] onto (0,10) . The answer is True. How??
Using the extreme value theorem, a continuous function $f$ from the closed interval $[0,1]$ onto (or even into) $(0,10)$ takes on a maximum $M$ and a minimum $m$ somewhere in $(0,10)$. But then any real in the interval $(0,m)$ is not taken on by $f$, so that $f$ is not onto. [Also $f$ cannot hit any real in $(M,10).$]
The extreme value theorem may be known by any calc 1 student, who might not know what "compactness" is.
For a function to be continuous the pre-image of an open set should be open. Here the pre-image of the open set $(0,10)$ is closed $[0,1]$. Hence the function is not continuous