W(x) = Σ aⁿ cos(bⁿπx) for n=0,1,2,… where 0<a<1 and ab>1+3π/2. Weierstrass (1872) shocked mathematicians by exhibiting a function that is continuous everywhere but differentiable nowhere — jagged at every scale. Zoom in to see self-similarity: it never "straightens out." Hölder exponent α = −log(a)/log(b).
Hölder exponent α = - ab = - (must be >1 for nowhere-differentiability) Zoom: scroll or drag
Drag to pan • Scroll to zoom • Condition ab > 1+3π/2 ≈ 10.42 gives strongest nowhere-differentiability guarantee