My question is almost the same as this question.
Let $f(\cdot, \cdot): R^2 \rightarrow R$. Suppose that:
(a) Fix any $y \in R, f(\cdot, y)$ is continuous almost everywhere.
(b) Fix any $x \in R, f(x, \cdot)$ is continuous almost everywhere.
Is $f(\cdot, \cdot): R^2 \rightarrow R$ is continuous almost everywhere?
I didn't really understand the reasoning in the comments of the linked question. Any help is most appreciated.