Monotonic non borel measurable function

Question

It is easy to see that if $I \subseteq \mathbb{R}$ is an interval, then every monotonic function $f: I \subseteq \mathbb{R} \rightarrow \mathbb{R}$ is Borel-measurable. Indeed, for every real number $\lambda$ the set $\{ x \in I : \lambda \leq f(x) \}$ is either an interval, a degenerate interval or the emptyset and therefore is a Borel set. Do there exist a Borel subset $B \in \mathfrak{B} (\mathbb{R})$ of $\mathbb{R}$ and a monotonic function $f: B \rightarrow \mathbb{R}$ such that $f$ is not Borel measurable?

Answer 1

Such a Borel subset $B$ can't exist as for $\lambda \in \mathbb R$, the inverse image of $[\lambda, \infty)$ under $f$ is the intersection of $B$ with either an interval, a degenerate interval or the empty set. It is therefore a Borel subset of the reals.