Weierstrass's Monster

Implementation, Code, and Audio Experiments

• Commenters share simple implementations of Weierstrass-type series in Python and TypeScript.
• The function’s graph looks like an audio waveform; people experiment with sonifying it and share YouTube examples.
• Observations: audio only reflects finitely many series terms (band-limited spectrum); one can swap the “audible” part with another signal (e.g., Chopin) while keeping a nowhere-differentiable function mathematically.

Counterexamples and Pedagogy

• Multiple “counterexamples in analysis/topology” books are recommended; Weierstrass appears prominently alongside many related constructions.
• Some praise these examples as essential for understanding why theorem hypotheses matter; others feel it can start to feel like “cheating.”
• There is discussion of rigorous proof culture, especially in French education, and how such counterexamples historically pushed math toward greater rigor.

Other Pathological Functions and Constructions

• Frequently mentioned: Dirichlet function, its “continuous only at 0” variant, Thomae’s (popcorn) function, Cantor function (Devil’s staircase), indicator functions of Cantor sets, Conway base-13 function, discrete metric, Schwarz lantern, staircase paradox, Gabriel’s horn.
• Several people note that “most” continuous functions are nowhere differentiable (via Baire category), making Weierstrass typical rather than exotic.

Computability, Measure, and Foundations

• One view: because non-computable reals dominate (full measure, uncountable), most wild examples are “nonexistent” computationally and of dubious practical value.
• Corrections: computable reals form a countable, measure-zero subset; all reals used in practice so far are computable.
• Long subthread debates Cantor’s diagonal argument, different infinities, density of rationals, separability, and whether future mathematics might reject some current axioms.
• Constructivist perspectives appear, questioning classical cardinality arguments but noting that Weierstrass itself is constructively well-behaved.

Intuition, Limits, and Geometry Paradoxes

• Several users discuss intuition-breaking examples: the staircase paradox, shapes with fixed perimeter but varying area, convergence modes (pointwise vs uniform vs weak*), and non-continuity of “perimeter” under limits.

Applications and Miscellany

• Brownian motion and related stochastic models (e.g., Langevin dynamics) are cited as real-world uses of nowhere-differentiable paths.
• Mentions of fractal interpolation functions, functions with first but not second derivatives, and questions about integrability of “random” functions and the role of the axiom of choice.