_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_.
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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!)