The use of Calculus for the analysis of real-world phenomenon depends entirely on our universe not only being continuous, but being differentiable.
By "real-world phenomenon" I mean things like the motion of a planet under the influence of a gravitational force (or just the action of forces upon objects in general), or fluid flow in a pipe. Just things that we can represent mathematically.
For every function modeling some real-life phenomenon, it seems that at a single point that function has a definite derivative, and doesn't just sharply turn.
In other words, we can consider almost any function modeling some real-world phenomenon locally linear.
It’s useful that we can: if small segments of a function are locally linear, then infinitely small segments become easy to analyze. We can then apply whatever properties we found to be true for the small segments of the function, to the entire function. That's what calculus is for.
But why is it that real life works this way - that movement, forces, etc are all differentiable functions? Or, more importantly than WHY this is (since I feel that "why" something it isn't really a question that can be discussed), are all real-world phenomenon differentiable?
Would a universe where things like (for example) the motion of a flower pot thrown from a plane and falling towards a planet due to gravity, would instead be modeled by a function that isn't differentiable, even be possible?