The Weierstrass Nowhere Differentiable Function
We construct a family of functions depending on two parameters that are everywhere continuous but nowhere differentiable. These functions are all lucunary Fourier series whose oscillation grows more rapidly than its modes decay. We use the Weierstrass M-Test to prove that the series converges to a continuous function. We carefully choose a sequence of values tending to zero on which the difference quotient grows exponentially.
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