Left continuous function which agrees with continuous function almost everywhere agrees everywhere

Question

Suppose we are working on $\mathbb{R}$ with the Lebesgue measure, and we have two functions $f$ and $g$, where $f$ is left continuous everywhere and $g$ is continuous everywhere. If we know they agree almost everywhere, can we say that they agree everywhere too?

Answer 1

Yes.

Hint: You need to show for any $a \in \mathbb{R}$, $f(a)=g(a)$. Show that in the interval $[a-1/n,a)$ you can find an $x_n$ such that $f(x_n)=g(x_n)$. Take limits.