Weierstrass functions

Weierstrass functions are mathematical constructs that demonstrate how a function can be continuous everywhere but not smooth or differentiable at any point. They are created as infinite sums of sine or cosine waves with carefully chosen amplitudes and frequencies, resulting in a jagged, complex shape that appears rough at all scales. These functions challenged earlier beliefs that continuous functions must be smooth, illustrating the rich and sometimes counterintuitive nature of mathematical analysis. Their significance lies in highlighting the complexity within seemingly simple notions of continuity and providing insights into fractal and chaotic structures.