Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$, and $g$ a continuous function. If $f(x)=g(x)$ for every $x$ in a dense set $A$ in $\mathbb{R}$, does $f$ having the Darboux property imply that $f$ and $g$ are equal on $\mathbb{R}$?
I know that if $f$ is continuous this is true, but are there any other weaker assumptions so that the statement is true (like having the Darboux property)?
For one, the Darboux property is not sufficient:
The Conway base 13 function $f$ has the Darboux property and is onto in every interval. In particular, we in each of the countably many intervals $I_n$ with rational endpoints some $\alpha_n\in I_n$ such that $f(0)=0$. Henc $f$ agrees with $g(x)=0$ on the dense set $\{\,\alpha_n\mid n\in\mathbb N\,\}$