Measurable function which is not equivalent to real continuous function

Question

Hi Can anyone help with the following problem:

1. Does there exist a measurable function $f:[0,1]\to \mathbb{R}$ which is not equivalent to any real continuous function on $[0,1]$?

2. Does there exist a non-measurable function $f:\mathbb{R}\to\mathbb{R}$ such that $f^{-1}(y)$ is measurable for any $y\in\mathbb{R}$.

Intuitively I believe that both of the answers are 'Yes, but I don't know how to prove?

Can anyone help. It would be highly appreciated.

Answer 1

The first one is pretty simple, the indicator function $\mathbf 1_{[0;1]}$ is not equivalent to any continuous function.

The second question is a bit more complicated, take $f:x\to x+\mathbf 1_{V}$ where $V$ is a non measurable subset of $[0;1]$. This function is not measurable (or else $f(x)-x=\mathbf 1_{V}$ would also be measurable) but $f^{-1}(y)$ has at most $2$ elements for every $y$, so it is measurable.