nowhere differentiable function

Intuition from calculus suggests a continuous curve is 'mostly smooth' — it might have a few sharp corners, but surely it has a tangent line almost everywhere. Weierstrass shattered that picture in 1872 with a function that is continuous at every point yet has a tangent line at NO point. The graph is so crinkled that, zoom in as you like, it never straightens out.

The classic construction is a series: W(x) = sum_{n>=0} a^n cos(b^n pi x), with 0 < a < 1, b an odd integer, and the product a b > 1 + 3 pi / 2. Because |a^n cos(b^n pi x)| <= a^n and sum a^n converges, the Weierstrass M-test makes the series converge uniformly, so by the uniform limit theorem W is continuous everywhere. But the high-frequency terms b^n pi x oscillate so violently that the difference quotients fail to converge at every single x, so W is differentiable nowhere.

This is the headline cautionary tale of the field: continuity does not imply differentiability, not even at one point, even though differentiability does imply continuity. It also shows uniform convergence preserves continuity but NOT differentiability — each partial sum is a smooth trigonometric polynomial, yet the uniform limit has no derivative anywhere. Such functions are not rare curiosities: in a precise topological sense, 'most' continuous functions are nowhere differentiable.