: Can the canonical theorem on Riemann integrability be taken as the definition of continuity? Can the canonical theorem on Riemann integrability (namely, that a bounded real-valued function defined on a closed interval is Riemann-integrable if, and only if, its discontinuities constitute a set of measure zero) be taken as the definition of continuity? If so, it would then be a theorem that a function f is continuous at a limit point a in its domain if, and only if, the limit of f(x) as x goes to a is f(a).

edit (2.Sep.2013, CST, MERCA)

I am not interested in defining continuity this way. I'm only interested in whether it is POSSIBLE to define continuity this way. That is, just as there is a (long) list of statements equivalent to the Axiom of Choice, so also was I wondering whether this is on the list of statements equivalent to the definition of continuity, that's all. I (weakly) conjecture that it is indeed on the list, and so I'm holding out for an answer that proves this. An answer that simply says that the answerer does not see how it can be done is not sufficient for me to give up my conjecture, but a counterexample certainly would.
The (Riemann)-Lebesgue Criterion gives a criterion for a bounded function to be almost everywhere continuous.  So you could take that as the definition of almost everywhere continuity for bounded functions if you want, although I don't see any particular advantage in doing so.  (Why do you want to define continuity in terms of Riemann integrability anyway?)

If $f$ is continuous at $c$ and we modify the value of $f$ at $c$ (while keeping every other value unchanged) then the modified function cannot be continuous at $c$.  The same holds if we modify the value of $f$ at any finite set of points on its domain.

On the other hand, if $f: [a,b] \rightarrow \mathbb{R}$ is Riemann integrable, then if we change the values of $f$ at any finite set we do not disturb the Riemann integrability or the value of the integral.  This is a nice elementary exercise involving the definition of the Riemann integral: the idea is to take an arbitrarily small subinterval around each modified point.

These diametrically opposite properties make it hard to see how one could possibly define continuity in terms of Riemann integrability.
You have given a definition of Riemann integrability for bounded functions, which states that the function f can be discontinuous on a set of measure 0.  So if it is discontinuous, it is discontinuous, even though it is perfectly integrable.  Your definition of continuity is correct, but there is no guarantee that f will meet that definition -- it might, or it might not.

However $\int f$ (i.e. any function F whose derivative is f where f is continuous) can be defined to be continuous although it will not be everywhere differentiable.

In general integration makes functions smoother, so that a slightly discontinuous function has a continuous integral F (in the sense of described above); a continuous function will yield an F that is differentiable and so on.