The Weierstrass function is a famous example of a function that is continuous everywhere but differentiable nowhere. It was introduced by Karl Weierstrass in the 19th century and serves as a key example in analysis and the study of pathological functions. The Weierstrass function demonstrates that continuity does not imply differentiability, challenging intuitive notions about smooth functions.
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A standard Weierstrass function is a uniformly convergent lacunary cosine series, here parameterized by and an integer frequency multiplier . Its continuity follows from the Weierstrass M-test. In the range , its modulus of continuity has the fractional-power scale for the positive odd-frequency model; the critical case can have the larger scale . These examples distinguish small approximation error from ordinary Lipschitz continuity.