Differentiability-Related Condition that Implies Continuity

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.)

• Does the condition "to have locally bounded partial derivatives" meets the requirement? – Etienne Jun 13 '14 at 19:54
• I'm not familiar with this property's consequences. Can you elaborate? – Optional Jun 14 '14 at 5:53
• This property implies continuity. This is proved in a similar (but simpler) way as the theorem "continuous partial derivatives implies differentiability". – Etienne Jun 14 '14 at 8:00