Weierstrass function

A continuous real-valued function that is differentiable at no point. Karl Weierstrass published an explicit example in 1872, challenging the assumption that continuous functions are usually smooth.

Weierstrass function formula: A series of cosine waves with geometrically shrinking amplitudes and increasing frequencies that defines a classic nowhere-differentiable function. The balance between shrinking amplitudes and growing frequencies creates persistent fine-scale oscillation.
Karl Weierstrass: A German mathematician whose work shaped nineteenth-century analysis, including an explicit continuous nowhere-differentiable function. He presented the counterexample that gave this function its enduring name.
Real-valued function: A function whose outputs are real numbers. The classic construction maps real inputs to real outputs.
Pathological function: A function that violates expectations formed from common or well-behaved examples. The Weierstrass function became a canonical example of a mathematically valid pathology.
Uniform convergence: A mode of convergence in which functions approach their limit uniformly across the entire domain. Uniform convergence of the defining series helps establish continuity of the function.
Bernhard Riemann: A German mathematician whose work transformed geometry, number theory, and mathematical analysis. A related nowhere-differentiable function had been attributed to Riemann before Weierstrass's example.
Continuity: A property of a function whose values change without jumps as its input varies. Continuity is the regularity the function retains at every point.
Baire category theorem: A theorem asserting that complete metric spaces cannot be expressed as countable unions of nowhere-dense sets. Related category arguments show that nowhere differentiability is widespread in spaces of continuous functions.
Hölder continuity: A regularity condition bounding changes in a function by a constant times a power of the input distance. The function has limited Hölder regularity despite being continuous everywhere.
Charles Cellérier: A Swiss mathematician who developed an explicit continuous nowhere-differentiable function before its later publication. His construction independently anticipated Weierstrass's famous example.

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