Let $f:[a,b]\rightarrow \mathbb{R}$ be a bounded, non constant function. We know that f is integrable in $[c,b]$ $\forall c \in [a,b]$. Prove that $f$ is Riemann integrable in $[a,b]$.
I have tried to prove it by saying that, as $f$ is integrable in $[c,b]$ $\forall c\in [a,b]$, I can take $c = a+ \varepsilon$, which implies that $f$ in integrable in $[a+\varepsilon,b]$; and since $\varepsilon$ is arbitrarily small, it's equivalent to saying that $f$ is integrable in $(a,b]$. Then, since adding $a$ to that set would give another point where $f$ is either continuous (which would mean that it's still integrable) or discontinuous (which would cause a finite number of discontinuities, 1, therefore making $f$ integrable), $f$ must be integrable in [a,b].
Is the reasoning correct?