Let $f: A \rightarrow \mathbb{R} $ be integrable and let $g=f$ except at fintely many points. Show that $g$ is integrable and that $\int_A f = \int_A g$
The question above is from Spivak's "Calculus on Manifolds" question 3-2 in the chapter on integration.
I want to prove this by using the theorem that states that a function is integrable iff $ \exists p $ such that $U(f,p)-L(f,p) < \epsilon$ $, \forall \epsilon$
Since I can assume that this condition holds for $f$ I want to find a partition $p'$ such that $U(g,p') \leq U(f,p)$ and $L(g,p') > L(f,p)$.
I have tried by starting with the case where $f$ and $g$ only differ at one point. It is then clear that in the case where $g(x_0) > f(x_0)$, that we can satisfy the inequality $U(g,p') \leq U(f,p)$ by refining the $p$ that satisfies $U(f,p)-L(f,p) < \epsilon$.
My Question How can I precisely pick $p'$ so that $U(g,p')=U(f,p)$ in the special case I described above? Intuitively, I can just keep refining $p$ and that partition which includes $x_0$ will tend to $0$, but I want to rigorously know at what point will $U(g,p')=U(f,p)$?
Thanks