Riemann integral question Suppose that $ f : [a,b] \rightarrow \mathbb{R}$ is Riemann integrable on $[a,b]$ and $g:[a,b] \rightarrow \mathbb{R}$ differs from $f$ at only one point $x_0 \in [a,b]$, that is, $g(x)=f(x)$ for $x \neq x_0$ and $g(x_0) \neq f(x_0)$. Show that $g$ is Riemann integrable on $[a,b]$.
I'm having a little trouble, my thing was that maybe find a partition and look at how it behaves in the partition containing $x_0$
Appreciate any help
 A: Here is an unnecessarily slick answer: 
There is a famous Lebesgue criterion for Riemann integrability of a function $f: [a,b] \rightarrow \mathbb{R}$ (my colleague Roy Smith informs me that it can actually be found already in the work of Riemann!): it is necessary and sufficient that $f$ be bounded and that its set of discontinuities have (Lebesgue!) measure zero.
Given this: it is an easy exercise to show that modifying a function by changing its values at any finite set $S$ does not change its boundedness/unboundedness, and similarly could only create or destroy discontinuities at $x$ for $x \in S$.  So the Lebesgue criterion applies here.   (Beware: changing a function at a countable set of values only can change the continuity at every point: I leave it to the reader to supply the canonical example of this.)
Of course one can -- and should -- also show this directly from the definition of Riemann integrability.
A: General hint: if you are faced with a problem, first try to understand what makes the problem interesting, and what is the noise; then filter the noise and extract the interesting part.
Let $\delta_z(x) = 1$ for $x = z$ and $\delta_z(x) = 0$ otherwise. If we showed that for any $z$ the integral $\int \delta_z(x) dx$ exists and equals $zero$ (this is the interesting part), we would have: $$g = f + c\delta_z(x)$$ and (this is the noise done by your profesor during the course) $$\int g(x) dx = \int f(x) + c\delta_z(x) dx = \int f(x) dx + c\int \delta_z(x) dx = \int f(x) dx$$
The interesting part is up to you.
