Pointwise limit of continuous functions whose graph is in a given closed set

Let $C\subseteq\mathbb R^2$ be a closed set with the property that for every $x\in\mathbb R$, there exists at least one $y\in\mathbb R$ such that $(x,y)\in C$.

Does there exist a function $f:\mathbb R\to\mathbb R$ of Baire class 1 such that $(x,f(x))\in C$ for every $x\in\mathbb R$?