= Solution
For a <monotone function>[increasing function] $F$, define its one-sided limits by
$$
F(x-)=\sup_{t<x}F(t),
\qquad
F(x+)=\inf_{t>x}F(t).
$$
They exist because of <monotone function>[monotonicity]. The function is discontinuous at $x$ only if the jump interval
$$
J_x=(F(x-),F(x+))
$$
is nonempty. By the <density of the rational numbers>, choose $q_x\in J_x\cap\mathbb Q$.
If $x<y$, monotonicity gives $F(x+)\leq F(y-)$, so $J_x$ and $J_y$ are disjoint. Consequently $q_x\ne q_y$, and $x\mapsto q_x$ injects the set of discontinuities into $\mathbb Q$. Since the rationals are countable by part (ii), the set of discontinuities is countable.
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