TITLE: Borel measurable function in measure theory
QUESTION [0 upvotes]: Let $\newcommand{\R}{\mathbb{R}}f:\R \to \R$ be defined by 
\begin{align}
f(x)=
\begin{cases}
x,                & \text{if}\; |x| < c\\
0,                & \text{if}\;  |x| \geq c
\end{cases}
\end{align}
Prove that $f$ is borel measurable.
I begin to solve it by take open interval in Borel on $\R$ and need to show the pre-image of this interval is in borel on $\R$ 
likethat 
$f^{-1}(c,\infty)= \R$ if $c<0$

REPLY [0 votes]: I assume that there is a misprint in the definition of $f$, i.e., I assume that $f(x) = \min\{x, 0\}$.
As you have already said, you have just to compute the pre-image of open half-lines, i.e. $f^{-1}((c, +\infty))$ for every $c\in\mathbb{R}$, and check that these are Borel sets.
You have that
$$
f^{-1}((c, +\infty)) =
\begin{cases}
\emptyset, & \text{if}\ c \geq 0,\\
(c, +\infty), & \text{if}\ c < 0\,,
\end{cases}
$$
so your function is Borel.
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