[jacobparker]

Right, but the reasons why are different from the halting problem. As stated in the OP, "the key assumption in this argument is continuity". I'm not sure what you mean by the outputs being "expected to be considered a continuous function". In both the halting problem and this case the outputs are discrete (e.g. binary: halts/does not halt, left/right, etc.)

There are other impossibilty results, e.g. the FLP theorem. Despite having computing-based interpretations they're not all related to the halting problem. :)

That being said, maybe it's possible to generalize the halting problem to a continuity in some reasonable way and get something like Buridan's principle. It's not obvious to me how that would be done but I'd be interested to see!

[Supermancho]

> this case the outputs are discrete (e.g. binary: halts/does not halt, left/right, etc.)

As you vary the inputs, the outputs change. This results in a continuity. There is a question of ranges, but this is handled by a type system (even if that type is a pointer, you run through all memory for your function inputs).