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Had to vote this up out of patriotism, but actually these ratings don't seem perfect. SIAM Journal on Mathematical Analysis is a 2, whereas Transactions of the AMS is a 1?
"Continuous function" has a special definition in reverse mathematics which does not seem to mentioned in Wikipedia yet, but which I allude to in my answer. For in-depth study of this subject one should get hold of Simpson's book Subsystems of Second Order Arithmetic.
I think that's basically the definition of continuity. Since every object in reverse math that belongs to the model must be represented by an element or a subset of $\mathbb N$, one cannot just use the set-theoretical definition of a function. You're right, a set being $\Sigma_1$ definable in the model does not imply the set is in the model, just that there is a $\Sigma_1$ formula with parameters from the model that defines the set.
