I'm given a theorem:
Let $f$ be Riemann-integrable on $[a, b]$. Then in $[a, b]$ it has infinitely many points of continuity.
Which is proven by using Riemann sums and constructing sequence of intervals $[a, b] \supset [a_1, b_1] \supset ...$ which converges to some point $c$ and then proving continuity from definition.
Now, there is second theorem:
Let $f$ be Riemann-integrable on $[a, b]$. Then in $[a, b]$ cardinality of set of points of continuity is equal to $\mathbb{R}$.
Where proof is stated as "Variation of previous one". How can I show that cardinality of set of points of continuity is equal to $\mathbb{R}$?