= Nowhere monotonicity of Brownian motion
Almost surely, no restriction of a <Brownian motion> path to an interval of positive length is a <monotone function>. On a fixed interval, monotonicity forces the independent centered <normal random variables> given by every dyadic subdivision's increments to have the same sign. The probability is at most $2^{1-2^m}$ for a subdivision into $2^m$ pieces. A countable union over rational intervals gives the simultaneous pathwise assertion.
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