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An almost sure event is a measurable event of probability one. Countable intersections of such events still have probability one, allowing a conclusion to hold simultaneously for countably many parameter choices. Further deterministic reasoning may then extend it to uncountably many parameters, as in Brownian Hölder regularity or nowhere monotonicity of Brownian motion. An uncountable intersection need not preserve probability one without such additional reasoning.

The Brownian first-passage subordinator has a countable dense set of jump levels almost surely, and is continuous at every other level. Its paths are finite nondecreasing functions, so on a bounded level interval there can be only finitely many jumps larger than any fixed positive size; a countable union proves countability. For density, suppose its path were continuous throughout an open level interval. It is strictly increasing, since the original continuous Brownian path takes different values at different passage times. Its continuous inverse on that interval would parametrize a positive-length time interval on which the Brownian path increases monotonically. This contradicts nowhere monotonicity of Brownian motion. Thus every level interval contains a jump. In particular dense discontinuities do not mean that the path is nowhere continuous: it is continuous at every level outside a countable random set.

The times of local maxima of a Brownian motion path are dense in the half-line almost surely, and every such local maximum is a strict local maximum. Density follows from nowhere monotonicity of Brownian motion and the extreme value theorem. Strictness follows by checking that maxima over separated rational compact time intervals are unequal; the increment across their gap supplies an independent nondegenerate normal distribution.

Use standard Brownian motion, started at $X_0=0$. Fix $0\le \alpha <\beta$, and divide $[\alpha ,\beta ]$ into $N$ equal subintervals. Its Brownian increments are independent centred normal random variables with strictly positive variance, so each is nonnegative with probability $1/2$. A nondecreasing path on the entire interval must have every one of these increments nonnegative. Hencefor every $N$, and therefore this probability is zero.

There are only countably many pairs of rational endpoints $0\le \alpha <\beta$. Take the union of the corresponding null events. Every interval of positive length, even one selected depending on the path, contains an interval with rational endpoints. Outside that single null event no such interval can be nondecreasing. ThusApplying the same argument to $-X$ also rules out nonincreasing intervals, the full nowhere monotonicity of Brownian motion property.

Fix rational $0<p<q$. Divide $[p,q]$ into $N$ equal intervals. Its Brownian increments are independent centered normal variables, so the probability that all $N$ are nonnegative is $2^{-N}$. A nondecreasing path would force this event for every $N$, hence its probability is zero. The same argument with nonpositive increments excludes a nonincreasing path.

There are countably many rational pairs $p<q$, so with probability one none of these intervals supports a monotone path. Every real interval $0<a<b<\infty$ contains such a rational subinterval. Monotonicity on the larger interval would imply monotonicity on that subinterval, a contradiction. Thus the nowhere monotonicity of Brownian motion assertion holds simultaneously:Both nondecreasing and nonincreasing behavior, including a constant path segment, are excluded.

First prove nowhere monotonicity of Brownian motion. On a fixed interval $[a,b]$ with rational endpoints and $a<b$, the $2^m$ increments over its equal subdivision are independent centered normal random variables. If the Brownian motion path were nondecreasing, all these increments would be nonnegative, an event with probability $2^{-2^m}$. Letting $m\to \infty$ gives probability zero. The same argument excludes nonincreasing paths. A countable union over rational intervals shows that, almost surely, no nontrivial interval supports a monotone function restriction of the path, since every such interval contains one with rational endpoints.

Work on this event and on the event of continuous paths. Inside any open interval $I\subset (0,\infty )$, choose two separated smaller intervals, the first to the left of the second. The first contains $r<s$ with $B_r<B_s$, because its restriction is not nonincreasing. The second contains $u<v$ with $B_u>B_v$, because its restriction is not nondecreasing. Thus $r<s<u<v$, all in $I$.

By the extreme value theorem, the path attains its maximum on $[r,v]$. This value exceeds $B_r$, since it is at least $B_s$, and exceeds $B_v$, since it is at least $B_u$. A maximizing time therefore lies in $(r,v)$ and is a local maximum of Brownian motion. Every open interval in the half-line contains a positive-time interval of this kind. Consequently the set of local maxima of Brownian motion is dense in $[0,\infty )$ almost surely.