Let a real measurable function $f$ map every open set to the whole real line. Is there always a restriction to a set of measure zero doing the same?

Given $f:\mathbb{R}\rightarrow\mathbb{R}$ mapping each non-empty open set to the whole real line, is there always a set $A$ of measure zero such that the function $g:\mathbb{R}\rightarrow\mathbb{R}$ defined as $g(x)=f(x)$ on $A$ and $0$ everywhere else also maps every non-empty open set to the whole real line?

Edit: My idea is to use the sigmoid function to convert it into the equivalent problem about functions with domain $(0,1)$ and work in binary. Consider the set $B \subseteq (0,1)$ defined as the union over all $n$ of values in $(0,1)$ whose $n$th to $n+k$th binary places are 0, for some $k$ depending on $n$. This strategy lets us restrict to a set of arbirtrarily small positive measure. Can this idea be extended by incorporating the actual $f$ to get a let us restrict to a set of measure $0$?