If two functions are both increasing and equal almost everywhere, do they have the same discontinuous points?

Question

Suppose that $f,g$ are both functions from $[0,1]\rightarrow[0,1]$.
$f$ and $g$ are both increasing.
$f=g$ almost everywhere.
Question: If $f$ and $g$ have the same discontinuous points?

My conjecture is yes. As I know, a monotonic function has countably many discontinuous points. If $f(x)$ is discontinuous at x=a but $g(x)$ is continuous at $a$, then around $a$ there must exist a small interval on which $f\neq g$. So $f$ and $g$ is not equal almost everywhere.

Is my conjecture correct?

Answer 1

You are correct. To add more detail to the argument:

Suppose $f$ is continuous at $c$ but $g$ is discontinuous there with value $g(c) = g(c+) = f(c).$ WLOG suppose $g(c+) > g(c-).$ There exists $\delta$ such that $f(x) > g(c-) \geqslant g(x)$ for all $x \in (c-\delta,c)$ contradicting $f = g$ a.e.