First, I don’t think it’s a good idea to have to rely on the axiom of choice in order to be able to define continuity.
Now, from my point of view, saying that continuity is defined in terms of limits is the wrong way to look at it. Continuity is a property relative to the topology of your space. If you define continuity in terms of open sets, I find that not only the definition does make sense, but also it extends in general to any topological space. But I kind of understand that not everyone will find this intuitive.
Also, I believe that your definitions that replace the limits in terms of hyperreals have to take into account all possible infinitesimals, and thus I don’t understand how it’s really any different that the sequential characterization of limits. But maybe I’m missing something.
Let \(X,Y\) be topological spaces. Then a function \(f:X\rightarrow Y\) is continuous if and only if whenever \((x_d)_{d\in D}\) is a net that converges to the point \(x\), the net \((f(x_d))_{d\in D}\) also converges to the point \(f(x)\). This is not very hard to prove. This means that we do not have to discuss as to whether continuity should be defined in terms of open sets instead of limits because both notions apply to all topological spaces. If anything, one should define continuity in terms of closed sets instead of open sets since closed generalize slightly better to objects known as closure systems (which are like topological spaces, but we do not require the union of two closed sets to be closed). For example, the collection of all subgroups of a group is a closure system, but the complements of the subgroups of a group have little importance, so if we want the definition that makes sense in the most general context, closed sets behave better than open sets. And as a bonus, the definition of continuity works well when we are taking the inverse image of closed sets and when we are taking the closure of the image of a set.
With that being said, the good thing about continuity is that it has enough characterizations so that at least one of these characterizations is satisfying (and general topology texts should give all of these characterizations even in the context of closure systems so that the reader can obtain such satisfaction with the characterization of his or her choosing).
First, I don’t think it’s a good idea to have to rely on the axiom of choice in order to be able to define continuity.
Now, from my point of view, saying that continuity is defined in terms of limits is the wrong way to look at it. Continuity is a property relative to the topology of your space. If you define continuity in terms of open sets, I find that not only the definition does make sense, but also it extends in general to any topological space. But I kind of understand that not everyone will find this intuitive.
Also, I believe that your definitions that replace the limits in terms of hyperreals have to take into account all possible infinitesimals, and thus I don’t understand how it’s really any different that the sequential characterization of limits. But maybe I’m missing something.
Let \(X,Y\) be topological spaces. Then a function \(f:X\rightarrow Y\) is continuous if and only if whenever \((x_d)_{d\in D}\) is a net that converges to the point \(x\), the net \((f(x_d))_{d\in D}\) also converges to the point \(f(x)\). This is not very hard to prove. This means that we do not have to discuss as to whether continuity should be defined in terms of open sets instead of limits because both notions apply to all topological spaces. If anything, one should define continuity in terms of closed sets instead of open sets since closed generalize slightly better to objects known as closure systems (which are like topological spaces, but we do not require the union of two closed sets to be closed). For example, the collection of all subgroups of a group is a closure system, but the complements of the subgroups of a group have little importance, so if we want the definition that makes sense in the most general context, closed sets behave better than open sets. And as a bonus, the definition of continuity works well when we are taking the inverse image of closed sets and when we are taking the closure of the image of a set.
With that being said, the good thing about continuity is that it has enough characterizations so that at least one of these characterizations is satisfying (and general topology texts should give all of these characterizations even in the context of closure systems so that the reader can obtain such satisfaction with the characterization of his or her choosing).