A deep dive into Weierstrass's famous construction: an infinite sum of cosines that stays continuous everywhere but has no tangent anywhere. We’ll unpack how shrinking amplitudes and rapidly increasing frequencies create endless jaggedness, trace the historical shock to 19th‑century intuition, compare with Riemann’s near-miss, and connect to fractals and modern analysis where such functions aren’t pathologies but natural in the landscape of continuous functions.

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