The Weierstrass Function - Continuous but Nowhere Differentiable

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Parameter a (amplitude): 0.5

Parameter b (frequency): 7

Terms: 50

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Fractal Dim. —

ab value —

The Weierstrass Function

W(x) = Σ aⁿcos(bⁿπx)

Sum from n=0 to ∞
0 < a < 1, b odd integer, ab > 1

⚠️ Pathological Properties

At EVERY point:
• Continuous ✓
• No derivative ✗
• No tangent line ✗

"An outrage against common sense."

— Henri Poincaré (1899)

😈 The "Monster"

In 1872, Karl Weierstrass presented a function that shocked mathematicians: it was continuous everywhere (you can draw it without lifting your pen) yet had no derivative at any point (no tangent line anywhere). This violated the prevailing belief that continuity implied smoothness "almost everywhere."

🔬 Why It Works

Each term aⁿcos(bⁿπx) oscillates faster and faster (frequency grows as bⁿ) but with smaller amplitude (shrinks as aⁿ). When ab > 1, the increasing oscillation outpaces the decreasing amplitude. Zoom in anywhere: you'll find infinitely many wiggles at every scale!

📐 Fractal Nature

The Weierstrass function is self-similar—zoom in and you see the same jagged pattern. Its fractal dimension D = 2 + ln(a)/ln(b), lying between 1 and 2. Not quite a line, not quite a surface. This was one of the first fractals studied, decades before Mandelbrot coined the term.

🌊 Modern Applications

Far from useless, the Weierstrass function now models: turbulent flow, stock price fluctuations, coastline irregularity, and signal processing. "Pathological" became typical—most continuous functions are nowhere differentiable!