TITLE: Continuity implies Borel-measurability?
QUESTION [6 upvotes]: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. Is it necessary that $f$ is Borel-measurable?
I'm considering $A=f^{-1}((a,\infty))$ where $a\in\mathbb{R}$. Is $A$ necessarily a Borel set? It looks like it should be, but I'm not sure.

REPLY [8 votes]: By definition of continuity, and since $(a, \infty)$ is open,
$$f^{-1}((a, \infty))$$ is an open subset of $\mathbb{R}$. Every open set is Borel.

REPLY [4 votes]: Hint: $(\alpha,\infty)$ is open, and $f$ is continuous.