I previously asked a related question here that I did not phrase as I intended. This is a revision of that question:
It is a well-known fact that differentiability implies continuity. And, for example, the existence and continuity of first partial derivatives implies continuity; but it also implies differentiability.
My question is this: is there some differentiability-related condition for a function that is both weaker than differentiability and stronger than continuity?
By "differentiability-related" I mean "involving the partial derivatives of various orders of the given function" (or any other conditions that you can think of that closely approximate whatever "differentiability-related" might mean, were it to be better defined.)