The intermediate value theorem states the following: Consider an interval $I=[a,b]$ in the real numbers $\mathbb{R}$ and a continuous function $f:I\to \mathbb{R}$. Then, If $u$ is a number between $f(a)$ and $f(b)$, $f(a)<u<f(b)$ (or $f(a)>u>f(b)$ ),

then there is a $c\in (a,b)$ such that $f(c)=u$.

What happens if we consider $\mathbb{R}$ instead of $I$ ?

Is there a simple method to prove the theorem in this case ? I know the method for $f:I\to \mathbb{R}$ with $I=[a,b]$, but here it is different.

then there is a $c\in (a,b)$ such that $f(c)=u$.

What happens if we consider $\mathbb{R}$ instead of $I$ ?

Is there a simple method to prove the theorem in this case ? I know the method for $f:I\to \mathbb{R}$ with $I=[a,b]$, but here it is different.