Function continuous and equal almost everywhere to a non continuous function. Lebesgue measure

Question

I need to find an example of a continuous function in $[0,1] \times [0,1]$ that is equal in almost everywhere to a function $g$ non-continuous in any point.

Answer 1

Take $f(x,y)=0$ and $g(x,y)=\mathbb{\chi}_{\mathbb{Q}^2\cap[0,1]^2}(x,y)$.

Then $f$ is continuous everywhere and equal to $g$ on $[0,1]^2\setminus\mathbb{Q}^2$ and since $\mathcal{L}^2(\mathbb{Q}^2)=0$ they are actually equal almost everywhere. Also $g$ is discontinuous everywhere.

Does that answer your question?