Weierstrass function

The Weierstrass function is a mathematical example of a function that is continuous everywhere but nowhere differentiable, meaning it has no smooth points. It is constructed by summing infinitely many oscillating sine or cosine waves with decreasing amplitudes and increasing frequencies. This creates a highly jagged, fractal-like curve that appears rough at any scale, showing how complex behavior can emerge from simple rules. The function challenges intuition by being perfectly continuous without any smooth slopes, illustrating the rich and often surprising properties of mathematical functions.