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.