Weierstrass Function

The Weierstrass function is a real-valued function that is continuous at every point yet differentiable at none of them, a jagged, self-similar curve that looks rough at every scale no matter how far one zooms in. Karl Weierstrass presented it to the Prussian Academy of Sciences on July 18, 1872, as the first published example of such a function. Before that, mathematicians including Gauss had generally assumed that a continuous function could fail to be differentiable only at a few isolated points, and Weierstrass's example overturned that assumption, unsettling several existing proofs that had leaned on unstated, geometrically intuitive notions of smoothness. Charles Hermite called functions of this kind a lamentable scourge, and the function was hard to visualize before computers existed, but it later found genuine use in modeling Brownian motion and other processes that behave in an infinitely jagged way.