TITLE: Continuous function and Borel sets
QUESTION [6 upvotes]: If $f:\mathbb{R}^p \rightarrow \mathbb{R}$ is continuous and $B\subset \mathbb{R}$ is Borel set, how to show that $f^{-1}(B)$ is also Borel set?
I was trying to construct a $\sigma$-ring of sets whose preimage is Borel, but they're not necessarily all borel in that sigma ring...
What would be the main idea for the proof here?

REPLY [6 votes]: Hint: prove that $\mathcal A := \{B \in B(\Bbb R): f^{-1}(B) \in B(\Bbb R^p)\}$ is a sigma algebra containing the open sets, where $B(\Bbb R^d)$ is the Borel sigma algebra on $\Bbb R^d$.
Added:
This way, $\tau \subset \mathcal A$ (where $\tau$ is the usual topology of $\Bbb R$), "applying $\sigma$ both sides" we get $\sigma(\tau)\subset \cal A$, therefore $B(\Bbb R)\subset \cal A$. Which means that each $B \in B(\Bbb R)$ is in $\cal A$, i.e. $f^{-1}(B) \in B(\Bbb R^p)$.