Given a continuous monotone function $f: \mathbb{R} \rightarrow \mathbb{R}$, is it true that for any point $x \in \mathbb{R}$, $f^{-1}(x)$ must also be connected?
The monotonicity is defined as a non strict monotonicity, and therefore $f^{-1}(x)$ is not necessarily a point.
A simmilar property has already been proven false for $\mathbb{R}^m$ in general in the question below, but I want to know if it is true for the case of $\mathbb{R}$, and in the case of a point $x$ insted of a subset $C \subset \mathbb{R}$.
Yes, it is true. Suppose otherwise. Since, in $\Bbb R$, being connected is the same thing as being an interval, if the statement was false, there would be $a,b\in f^{-1}(x)$, with $a<b$, and there would be $c\in(a,b)$ with $c\notin f^{-1}(x)$. But then $f(c)>x$ or $f(c)<x$. Since $f(a)=f(b)=x$, this contradicts the assumption that $f$ is monotonous.
Note that, as you were told in the comments, continuity is not needed to prove this.