Let f a Riemann integrable function (in the proper sense) over [a,b]. Let g be another function that is different from f only in a finite number of points.
How can I show that g is Riemann integrable and that the two integrals (of f and g) over [a,b] are equal?
Thank you!
Let define the functions $\chi_a$ as
$$\chi_a(x) = \begin{cases} 1 & \text{ if } x=a \\ 0 & \text { otherwise }\end{cases}$$
We have that $\chi_a$ is riemann integrable and $\int\chi_a dx = 0$. Now we have that $g-f$ is non-zero in finite many points which means we can write it as a finite sum:
$$g(x)-f(x) = \sum c_a\chi_a$$
A finite sum of rimann integrable functions is Riemann integrable and it's integral is the sum of the integrals of the terms. So $\int g(x)-f(x) dx = 0$ and also $g(x) = f(x) + g(x)-f(x)$ means that $g(x)$ is Riemann integrable and it's integral is:
$$\int g(x) dx = \int f(x) dx + \int(g(x) - f(x))dx = \int f(x) dx + 0$$