Showing that $f$ is not measurable I'm learning about measure theory, specifically measurable functions, and need help with the following problem:

Let $N$ be a non-measurable subset of $[0, 1]$ and define
$$f(x) = \begin{cases} x, & x \in N  \\ -x, & x \in [0,1] \setminus N \end{cases}$$
$(1)$ Show that $f(x)$ is a non-measurable function.
$(2)$ Show that $\forall a \in \mathbb R$ the set $\{f(x) = a\}$ is measurable.

This problem looks rather simple but I think I'm missing some key points. I tried to prove $(1)$ by contradiction, that is assuming $f$ is measurable but I didn't get far. For $(2)$ my guess is that $m(\{f(x) = a\}) = 0$ (since by definition the Lebesgue measure of a single point is $0$) but I don't know how to show this properly.

Hint: $f$ is not measurable since $f^{-1}((0,1))=N-\{0\}$.
