There can be no $\vec\forall\vec\exists$ definition of continuity, since continuity is not preserved to limits of chains of models.
Specifically, start with the real field, and build an elementary tower of hyperreal models over it, each with more infinitesimals wrt to the previous.
Let f_n be a continuous function in the nth model, mostly zero except for n narrow bumps up to 1 close to zero. Each next function adds another bump in the new infinitesimal region near zero, and coheres with the previous function. All have f_n(0)=0.
Each f_n is continuous in the nth model, but the limit model does not think f is continuous, since it will have bumps jumping up to 1 near 0 but f(0)=0.
In short, you can always add one more big bump near zero while staying continuous, but the limit model will not think the limit f is continuous, since it has those jumps up to 1 arbitrarily close to 0.
So the property of continuity is not preserved by unions of chains, and so it cannot be characterized by a $\vec\forall\vec\exists$ property, since these always are preserved to limits of chains.
(I will edit later, currently on my phone at a bar, sorry.)