Let $f:[a, b] \rightarrow \mathbb{R}$ be a bounded function. The usual way of defining the Riemann integral is to take lower sums and upper sums of partitions of $[a, b]$ and going from there.
However, I stumbled upon the following question: The lower sums form a bounded set, since it is bounded from above by any upper sum and bounded from below by $C(b-a)$, where $C$ is a lower bound for $f$ on $[a, b]$. Is this set connected? In other words, given any real number $c$ between the bounds of the lower sum, can you take a partition $P$ such that the lower sum of $f$ with respect to $P$ is $c$?
My intuition tells me that this would hold for a continuous function, but I couldn't even prove it for the identity function on $[0, 1]$. Is my intuition correct? If it is, can we relax any conditions on $f$? I would really like to work on this myself, so any hints would be appreciated.