Continuous image of $(0, 1)$ equal to half-closed interval

Question

I am looking for an example of a continuous map $f: \mathbb{R} \to \mathbb{R}$ which maps $(0, 1)$ to some $(a, b]$.

I also would like to find such a map where the image of $(0, 1)$ is instead some $[a, b)$.

I know that the image must be an interval, and I see no reason why such maps could not exist.

But while $(-1, 1)$ admits an easy example (e.g., take $f(x)={x^2}$), I can't find one for $(0,1)$.

Any thoughts?

Answer 1

What about a shifted parabola such as $$ y=(x-1/2)^2$$ for the first part and its reflection $$ y=-(x-1/2)^2$$ for the second question?

Of course you may change the power to any even positive integer as well.