I'll preface this by saying I don't have an advanced background in abstract algebra, so I'm sorry if these concepts already exist but I just have never heard of them.
So, in my study of real analysis, I remember in a course where we extended from the concept of a Gateaux derivative to the concept of a Fréchet derivative, because not only did we seek a definition of the derivative that was defined over an infinite dimensional Banach space where the Gateaux derivative failed, but because we also wanted this new definition to imply continuity as the regular derivative does over differentiable functions, which is to say, if the Fréchet derivative of a linear operator exists, then that operator is also continuous.
Recently as I talked about in another question, I happened across a different type of abstraction of the differential operator, being a "derivation" that distills the differential operator down to its key properties of linearity and the product rule for its study in the context of abstract algebra.
My question is, if we are given the assumption that the derivation operation is defined over a general field, like a differential field or possibly a differential ring, then, is the derivation map also "continuous" in any abstract sense? Is there an abstract form of continuity that then implies the existence of limit-like objects just by these few abstract properties?
EDIT: I should add that if you can think of any caveats for such a condition, I'm not opposed to that. For instance, maybe such a differential field is limited to being "complete", and if so, what does this equate to in abstract algebra terms? Is this perhaps an infinite set of polynomials?