real analysis - Measure of big discontinuities

Let $D\subset\left[ 0,1\right] $ be a dense set, and $\mu$ Lebesgue
measure on $\left[ 0,1\right] .$

Suppose $f:\left[ 0,1\right] \rightarrow\left[ 0,1\right] $ is continuous
at each point in $D.$ Let $\overline{G}$ be the closure of the graph of $f$ on $\left[
0,1\right] ^{2}.$

Is it true that $\mu\left\{ x_{0}:\overline{G}\cap\left\{ x=x_{0}\right\}
\text{ is infinite}\right\} =0?$

Answer

Here's a counterexample: Let $D$ be the complement of a fat Cantor set $C \subseteq [0, 1]$ so $D$ is dense. Construct $f:[0, 1] \rightarrow [0, 1]$ such that $f$ is zero on $D$ and the graph of $f$ is dense in $C \times [0, 1]$.