Jordan measurability of "Area under the curve" of a function

Question

Let $I = [a,b] \subset \mathbb{R}$. Let $f : I \to \mathbb{R}$ be a riemann integrable function on $I$. Consider the set $S_f$ given by, $$S_f = \{ (x,y) \in \mathbb{R}^2 | x \in I, 0 \leq y \leq f(x) \}$$ I want to prove that $S_f$ is a jordan measurable subset of $\mathbb{R}^2$. I proved the result under the assumption that $f$ is continuous using the uniform continuity (and the fact that a set $X$ is jordan measurable iff its boundary $\partial X$ has measure $0$). But with Riemann integrability instead of continuity, I no more have the nice bounds that uniform continuity gives to contain the boundary in rectangles with arbitrarily small total area. Any help/solution is highly appreciated. Thanks in advanced.