3 ms·

Here’s a really cool thing about continuous-everywhere differentiable-nowhere functions. A consequence of the Baire Category Theorem is that most continuous fun

Here’s a really cool thing about continuous-everywhere differentiable-nowhere functions. A consequence of the Baire Category Theorem is that most continuous functions are nowhere-differentiable (in the same sense that “most” real numbers are irrational). The familiar functions from calculus are the vanishing minority of continuous functions.

coke12 2y ago
Are there any hierarchies in mathematics that aren't like this? It seems like most hierarchies are constructed such that going up one "level" expands your scope so much that the previous rung looks like a drop of water in the ocean. Examples include, for example: the Chomsky hierarchy of languages, where most context-free languages are not regular; Turing computability/solvability; and so on.