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: 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.
A: I mean, math is all about building a general, rigorous framework, right? If we took a theorem that typically requires differentiability and changed it to just require continuity, yeah it might work most of the time, but it's not true in general; there will be counter-examples. That is, the result is not true. So, what's the point? Why do we want something that is only true sometimes, and how would this theory translate to application? 
Differentiability is a stronger requirement than continuity that is needed for many, many results. Sometimes, we need quite a bit more! What if I want to guarantee that a function has a local inverse? What if we want to approximate a function by nicer functions? If we only paid attention to continuous functions and nothing stronger, our mathematics wouldn't be nearly as rich. 
EDIT: On a second read-through of the original post, I see that I misread it a bit the first time. The post was, more than anything, asking why we need differentiability in results, and why continuity is not enough. So, my first paragraph isn't really as important as the second. The key takeaway is this: continuity is not an overly strong condition, and we often need more to get "nice" results. A lot of results, obviously or otherwise, need the function to be fairly well-behaved, and continuous functions can behavior more wildly than differentiable functions in many ways. In many cases, the smoother, the better!
