Is it possible to have $f: [a,b] \to \mathbb{R}$ and $f \in R[c,b]$ for $c\in (a,b)$ and $f \notin R[a,b]$ ? If so, what kind of function behavior are we talking about here? Clearly it's not continuous and not a step function. I would guess it would have to be something special like
$$f(x) := \frac{\sin(\cos(1/x))}{\sin(\cos(1/x))} \, \, \, \forall x \neq0 \\f(x): = 0 \, \, \, \text{when } x = 0 $$
Where the function is becoming more discontinuous as $x$ gets closer to zero but can still be bounded by 1 for infinitely many nonzero lengthed intervals.