Must $f$ be measurable if each $f^{-1}(c)$ is? Suppose $f$ is a real-valued function on $\mathbb R$ such that $f^{−1}(c)$ is measurable for each number $c$. 
Is $f$ necessarily measurable?

Consider for example a set $E \subset [0,1]$; then the function
$$
f(x)=\begin{cases}
x & x \in E\\
-x & x \in [0,1]\setminus E
\end{cases}
$$
is measurable if and only if $E$ is measurable. On the other hand, $f$ is injective, hence $f^{-1}(c)$ is either empty, either a singleton (in both cases, anyway, $f^{-1}(c)$ is measurable). 
