If $f:\mathbb{R}\to \mathbb{R}$ is a function such that:
$$f(x)=\{\begin{array}{ll}1,& \text{if }x>0\\ 0,& \text{if }x=0\\ -1,& \text{if }x<0\end{array}$$
then $f(x)$ is:

1. Many-one: Consider $x_1=1$ and $x_2=2$. Since both are $>0$, $f(1)=1$ and $f(2)=1$. Since different inputs give the same output, the function is many-one.2. Into/Onto: The codomain is given as $\mathbb{R}$ (all real numbers). However, the range (the set of actual outputs) is only $\{-1,0,1\}$.Since $\text{Range}\ne \text{Codomain}$, the function is into.Therefore, $f(x)$ is many-one and into.