Is it true that there is a continuous, onto map from $(0,1]$ to $\mathbb R$?

Question

Is it true that there is a continuous, onto map from $(0,1]$ to $\mathbb R$?

I am trying but could not approach.

Answer 1

One example is $$ f(x) = \frac{\sin(1/x)}{x} $$

Answer 2

Try $\frac{\sin(\frac{1}{x})}{x}$.

Answer 3

$f(x) = \frac{1}{x}$ maps $(0,1]$ to $[1,+\infty)$. Can you map the latter onto $\Bbb R$ continuously? Then compose.