 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.