Does there exist a continuous onto function from an open interval to a semi open interval?

Question

Let $f$ is a function from $X$ to $Y$. Here $X= (0,1)$ and $Y=(0,1]$. Can $f$ be continuous and onto? I am a bit confused. Please give me some hint.

Answer 1

Yes sure, consider the function $f(x) = 4(x-x^2)$, it is continuous and onto.
Note that $f(0.5) = 4(0.25) = 1$.

Answer 2

Yes, first map $(0,1)\to(0,2):x\mapsto 2x$, then shift $(0,2)\to(-1,1):x\mapsto x-1$, then reflect the left half onto the right half by $(-1,1)\to [0,1):x\mapsto x^2$, and finally reflect in $1/2$ to get the closed end at 1 $[0,1)\to(0,1]:x\mapsto 1-x$. Now let $f$ be the composition of these functions.