Is there a function such that when we change its values on $\mathbb{Q}$ its set of discontinuities is exactly $\mathbb{Q}$?

Question

Given a continuous function $\mathbb{R}\rightarrow \mathbb{R}$, when we change its value at a finite set $D$ of points, we get a function whose set of discontinuities is exactly $D$. However this is not true in general when $D$ is infinite. For example when $D=\mathbb{Q}$, is there a continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that there exists another function $\tilde{f}:\mathbb{R}\rightarrow \mathbb{R}$ that differs from $f$ exactly at points in $\mathbb{Q}$, such that the set of discontinuities of $\tilde{f}$ is exactly $\mathbb{Q}$?