It is easy to take for granted the simple idea that similar input $x$ to a function $f(x)$ should yield similar outputs - such that if the difference between $x$ is arbitrary small, then we should also be able to get arbitrary small changes in $f(x)$.
But what is this fundamental property called? Is it just function continuity? But even continuity seems to rely on this fact. What stops me from arguing that only arbitrary far-apart numbers as inputs will yield similar outputs?
It feels like there should exist an even more fundamental axiom upon which all this is based. What is this axiom called?
This is not an axiom, it's exactly what the property "continuity" means. And there are functions which aren't continuous: for an extreme example, take Conway's base 13 function $h$, which has the property that for every $x$ and $y$, and for every $\delta>0$, there is some $c\in (x-\delta, x+\delta)$ such that $h(c)=y$. That is, it's completely uncontrollable.
Regarding axioms: There are certain (extremely rare) contexts where it can make sense to work in a system one of whose axioms is "every function is continuous." Separately, one might argue that "every function occuring in nature is continuous," but that is imprecise. In general, though, if you're looking for an axiom that tells you that you can take continuity for granted, there isn't one - because you can't!