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Pathological Functions

A gallery of intuition-breaking functions: continuous but nowhere differentiable, flat yet climbing.

1. Length Zero, Points Uncountable — 4. Topology
2. The Cantor Function — 4. Topology
3. The Weierstrass Function — 9. Differentiation
4. Volterra's Function — 10. Integration
5. Functions of Bounded Variation — 10. Integration
6. Jordan Decomposition — 10. Integration
7. Darboux's Theorem — 9. Differentiation
8. Weierstrass Approximation Theorem — 5. Approximation
9. When the Limit Breaks — 3. Continuity