What is the use or importance of continuity and differentiability? In a lot of mathematical proofs I often see things like "Assume $f$ is continuous" or "Assume $f$ is differentiable" and sometimes I've even seen both "Assume $f$ is continuous and differentiable", though I believe (differentiability implies/requires continuity).
What is the use or utility of this? Under what kind of circumstances, going into a problem or framework, would we want something to be continuous or differentiable? Continuous I can understand as a kind of "useful for things that don't have sudden jumps out of nowhere" but when would we want something to be differentiable as well?
For instance when reading about lots of probability curves I often see that these curves are defined up front as both continuous and differentiable. Why? What pushes us to start off with these definitions? What's the motivation? What do we "lose" if we do away with these assumptions?

A function being differentiable is important in a lot of analysis as a lot of theorems (such as Rolle's theorem, for instance) simply don't hold when a function is not differentiable. It's basically so that we can assume the function is "well-behaved" in a way so that we can apply standard theorems to the function; otherwise, some of these theorems aren't applicable anymore. Also, you can still have continuous curves which have some strange properties that we may want to avoid; for example, we may not want functions to have sharp (but still continuous) corners.
