 Solution

For a increasing function $F$, define its one-sided limits byThey exist because of monotonicity. The function is discontinuous at $x$ only if the jump intervalis nonempty. By the density of the rational numbers, choose $q_x\in J_x\cap \mathbb{Q}$.

If $x<y$, monotonicity gives $F(x+)\le F(y-)$, 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.