The 19th century view was that a function was analytic. Of course, some functions were less smooth than that, but analyticity was the starting assumption.
The 20th century view is that a function is continuous. Of course, some functions are smoother than that, but continuity is the starting assumption. (Have you ever seen a merely continuous function? I think there is one main example: a random Brownian path.)
The 20th century view has utterly taken over among pure mathematicians. One can understand this on the basis that analytic functions are too easy to offer research challenges. Unfortunately, the same view has been copied by the numerical analysts, who ought to have been guided by what applications look like. One sees this in their edifice of results expressed in the language of Sobolev spaces. The point of Sobolev spaces is to make precise distinctions between, say, a function with one derivative and a function with one-and-a-half derivatives. Nineteenth-century mathematicians did not spend their time on such distinctions.
And so it is that numerical analysts prove technical, rigorous theorems about the convergence of algorithms whose convergence is nowhere near optimal for most applied problems. Meanwhile pure mathematics continues its drift away from science.
[1 December 2018]