Continuous function from $(0, 1]$ onto $(0, 1)$?

Question

Is there any continuous function defined on $(0, 1]$ whose range is $(0, 1)$?

Answer 1

Something like $$x\mapsto \frac{1+(1-x)\sin(1/x)}{2} $$ will do.

Answer 2

Yes, there is. First send $(0,1]$ onto a spiral that winds inside a circle (e.g centered at $x=1/2$ and radius $1/2$) and then project it onto $(0,1)$.

Answer 3

$$f(x)=\frac12+\frac1\pi\arctan\left(\frac1x\cos\left(\frac1x\right)\right)$$

Answer 4

**Answer without words! see the figure **