I am confused regarding Riemann-integrability of functions with countable-many discontinuities.
Let's say we have one Riemann-integrable function $f:[a,b]\to \mathbb{R}$ with countably-infinite-many discontinuities and another function $g:[c,d]\to\mathbb{R}$ which is continuous (and hence Riemann-integrable).
Now suppose that we change successively countably-infinite-many points of $g$ such that $g$ becomes discontinuous at those points.
1.) I am wondering if the following two thoughts on proving the Riemann-integrability are correct:
-I can show the Riemann-integrability of $g$ via induction.
-Proving Riemann-integrability of $f$ via induction is not possible.
2.) If the Riemann-integrability of $f$ can't be proven by induction, why is that so?