*Presume* and *Imply* I am not sure about the usage of the word presume.
For example, is the sentence differentiability implies continuity equivalent to differentiability presumes continuity or to continuity presumes differentiability
or to none of them? How come?
 A: Presuming is something that people do. Author $X$ can presume something in order to take a particular approach in writing a proof. Conditions like differentiability and continuity do not presume anything about each other. I think you are really thinking about sufficient and necessary conditions. 
Differentiability implies continuity: this means that for differentiability to hold, continuity must also hold.
That is interchangeable with:
Differentiability necessitates (requires) continuity. 

To explain what I meant with sufficient and necessary conditions:
Say X is sufficient for Y. This is equivalent to If X, then Y. 
Note that this does not mean If Y, then X. The implication is one-way.
On the other hand, a necessary condition works like this:
Say A is necessary for B. This is equivalent to If B, then A. (Whenever we have B, we must have A --- it's necessary!)
A: "Differentiability presumes continuity" is legitimate English, but it is short for a slightly different statement: "To even talk about differentiability involves presuming continuity, because it cannot occur otherwise." This is true because "Differentiability implies continuity" is true, but it's about human discussion, not only about mathematical properties.
Compare the google hits for "suffering presumes".  It is never the suffering that makes the presumption, even when it is syntactically the subject of the verb, but the sentences work (for humans -- maybe not for NLP programs).
