An old problem: to show that *every bounded left integrable function is also Riemann integrable*.  

I know some (for me) not elementary proofs: [Gillespie](http://www.jstor.org/stable/2007121) and [Kristensen et al](http://www.jstor.org/stable/2311188) (theorem 1).  

The question: can it be an undergraduate problem, i.e. does an elementary proof exist ?



Cauchy's left endpoint integral (1823)
The following argument may not be the most direct one, but follows as a quick consequence of the characterization of Riemann integrability, which is perhaps the main result of the theory. 


For a bounded function $f:[a,b]\to\mathbb{R}$ and for $\lambda >0$ let's denote
$$J \_ \lambda :=\{ x\in [a,b]\, : \,\limsup \_{y\to x}f(y)-\liminf \_{y\to x}f(y) \ge \lambda\}\,.$$
The key point is that for any $\epsilon > 0$ there are two partitions $P^\*$ and $P \_ \*$ of the interval $[a,b]$, both with mesh less than $\epsilon$, such that the corresponding left sums $s\_L(f,P^\*)$ and $s\_L(f,P \_ \* )$ differ for more than $\lambda\operatorname{meas} (J \_ \lambda)-\epsilon$ (to construct $P^\*$ and $P \_ \*$, start from a covering of $J \_ \lambda$ by a finite collection of disjoint intervals of length less than $\epsilon$, and whose left endpoints belongs to $J\_\lambda$) . As a consequence
$$\limsup \_ {|P|\to0} s\_L(f,P) - \liminf \_ {|P|\to0} s\_L(f,P) \ge \lambda\operatorname{meas} (J \_ \lambda)\, , $$
and we conclude that if $f$ is left integrable in the Cauchy sense, then $\operatorname{meas} (J \_ \lambda)=0$ for all $\lambda >0$, that is, it is continuous a.e., hence Riemann integrable by the characterization. 


