Example of borel measurable function for which there doesnot exist a borel set such that function restricted on it is continuous

Question

Give an example of a Borel measurable function f from R to R such that there does not exist a set B ⊂ R such that $| R \setminus B |$ = 0 and $f|_B$ is a continuous function on B.
By Lusin's theorem there always exist a closed set F such that $f|_F$ is continuous and $|R\setminus F| <\epsilon$. So if $F_{n}$ is chosen in such a way that $|R\setminus F| < 1/n$ and B is taken to be the union of all such sets, then $| R \setminus B |$ = 0 .So the question narrows down to finding an example where it is continuous on each $F_{n}$ but not on its union.