Weierstrass Function

Key idea

A continuous function nowhere differentiable.

In depth

W(x) = Σ aⁿ cos(bⁿπx), with 0 < a < 1 and ab > 1 + 3π/2, is continuous everywhere but differentiable nowhere. It shattered the 19th-century belief that continuous functions are mostly smooth.

Worked example

Zooming into the graph shows the same jagged detail at every scale, as in a fractal.

Real-world uses

• Showing that some continuous curves have no tangent at any point
• Models of rough surfaces

Common mistake

Thinking such a function cannot be defined by a formula.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.