Changing Finitely/Countably many points affect Riemann Integral

Question

Is it true that the Riemann integral does not change if you were to change the function values at finitely many points? What about for countable many points changed, does it still hold?

Edit: Would prefer formal proofs for both

Answer 1

No, for constant function $1$ on $[0,1]$, if you change the function taking rational numbers on $[0,1]$ with values $0$, that is, to be $\chi_{{\bf{Q}^{c}}\cap[0,1]}$, then the new function is not Riemann integrable.

For finitely many, it does not affect the value. But for countably many case, if it were given that the new function is still Riemann integrable, then the value does not change. The reasoning is the following:

Given that both $f$ and $g$ are Riemann integrable but $g$ differ from $f$ countably many points, then it is still $f=g$ a.e. And we know that Riemann integrability implies the Lebesgue one with the same value, and Lebesgue integrals do not matter for a.e. difference, so $(R)\displaystyle\int f=(L)\int f=(L)\int g=(R)\int g$.