TITLE: If $f$ is of bounded variation is $f$ Riemann integrable?
QUESTION [15 upvotes]: I want to know if $f$ is of bounded variation on $[a,b]$ does it follow that $f$ is Riemann integrable on $[a,b]$?

REPLY [6 votes]: I don't understand why people always prefer the solution with the representation of a BV function as a difference of two monotone functions. This is of course correct but it needs additional work. A proof with the definition of integrability, in my opinion, is way more convenient and self-contained without using any 'black boxes'. The weird thing is that this proof cannot be found anywhere on the internet as if someone really needs to quote the aforementioned representation...
Let $\varepsilon>0$ and $S(P),$ $s(P)$ be the upper and the lower sum respectively of the your function with respect to the partition $P=\{a=x_0<\ldots<x_n=b\}$ with $x_j-x_{j-1}=\frac{b-a}{n}$ (we will choose this $n$ later). Then, we have that 
$$S(P)-s(P)=\sum_{j=1}^n (M_j-m_j)(x_j-x_{j-1})=\frac{b-a}{n}\cdot\sum_{j=1}^n (M_j-m_j),$$ where $M_j,$ $m_j$ is the sup and the inf of the function at the corresponding subinterval $[x_{j-1},x_j]$ respectively. Pick $c_j, c_j'\in [x_{j-1},x_j]$ with $f(c_j)>M_j-1/n$ and $f(c_j')<m_j+1/n.$ Then, if $K$ is the BV constant, we have that 
$$S(P)-s(P)\leq \frac{b-a}{n}\cdot\sum_{j=1}^n \left( \left|f(c_j)-f(c_j')\right| +2/n\right)<\frac{b-a}{n}\cdot(K+2)<\varepsilon$$ for the appropriate $n.$ We are done by the characterization of the definition of the Riemann integrability.