Consider the function:

$f:\mathbb{R}\to \mathbb{R}$

$f(x)=\frac{x}{|x|+1}$

I already showed that this function is continuous and injective. Maybe this could help? Maybe we can use the intermediate value theorem? The problem is that I don't have an interval [a,b] with a,b $\in $ $\mathbb{R}$. I know that I can consider the limit, but we didn't proved the fact, that I can use the intermediate value theorem for limits.

$f:\mathbb{R}\to \mathbb{R}$

$f(x)=\frac{x}{|x|+1}$

I already showed that this function is continuous and injective. Maybe this could help? Maybe we can use the intermediate value theorem? The problem is that I don't have an interval [a,b] with a,b $\in $ $\mathbb{R}$. I know that I can consider the limit, but we didn't proved the fact, that I can use the intermediate value theorem for limits.