Weierstrass' nondifferentiable function
Weierstrass' nondifferentiable function is the earliest known example of a pathological function – a function that gives rise to a pathological curve. It was investigated by Karl Weierstrass, but had been first discovered by Bernhard Riemann, and is defined as:
where 0 < a < 1, b is a positive odd integer, and ab > 1 + 3π/2.
The Weierstrass function is everywhere continuous but nowhere differentiable; in other words, no tangent exists to its curve at any point. Constructed from an infinite sum of trigonometric functions, it is the densely-nested oscillating structure that makes the definition of a tangent line impossible.