Brownian paths are nowhere differentiable

Suppose the continuous path $f:=B(\omega )$ has a finite two-sided derivative at some $s\in (a,b)$, or a finite right derivative at $s=a$, or a finite left derivative at $s=b$, with absolute value at most $C$; use both sides for an interior point, the right side at a and the left side at b, applying [F2] to obtain $\delta >0$ such that $\mathrm{\mid }f(t)-f(s)\mathrm{\mid }\le (C+1)\mathrm{\mid }t-s\mathrm{\mid }$ for every $t\in [a,b]$ on the permitted side or sides of $s$ with $0<\mathrm{\mid }t-s\mathrm{\mid }<\delta$.

Fix $n\ge 6$ with $(b-a)\mathrm{/}n<\delta \mathrm{/}6$, write $h:=(b-a)\mathrm{/}n$ and $t_k:=a+kh$, and put $k_0:=\max \{k\in \{0,\dots ,n-1\}:t_k\le s\}$. Then $t_{k_0}\le s\le t_{k_0+1}$, including $k_0=n-1$ when $s=b$. If $k_0+5\le n$ take the block of five increments beginning at $t_{k_0}$, and otherwise take the block of five increments ending at $t_n$. In either case all endpoints lie in $[a,b]$, on the side of $s$ allowed in step 1.1 for the endpoint cases, and within distance $6h<\delta$ of $s$, so each increment has absolute value at most $2(C+1)\cdot 6h=12(C+1)h=:D\mathrm{/}n$ with $D:=12(C+1)(b-a)$.

For n=0,1,2,3,4,5 set G_n to be the empty event, without defining h or a mesh for those indices. For integers n>=6 define $G_n$ to be the event that some block of five consecutive increments $B_{t_{k+i}}-B_{t_{k+i-1}}$ ($k=0,\dots ,n-5$, $i=1,\dots ,5$) has all five absolute values at most $D\mathrm{/}n$, where $h=(b-a)\mathrm{/}n$. Every G_n is a finite union of finite intersections of measurable coordinate events. Insert 0 before a if a>0 and use the subfamily of grid increments in [a,b]; the increments of one block are independent with laws $N(0,h)$ by [F1], so by [F3] and independence the probability for a fixed block is at most $y_n^5$, where $y_n:=D\mathrm{/}\sqrt{(b-a)n}=12(C+1)\sqrt{(b-a)\mathrm{/}n}\le 1$ for large $n$. Hence $P(G_n)\le n{y}_n^5={12}^5(C+1{)}^5(b-a{)}^{5\mathrm{/}2}{n}^{-3\mathrm{/}2}$, and ${\sum }_{n}P(G_n)<\mathrm{\infty }$: the finitely many remaining initial terms are at most one each, and ${\sum }_{n=2^j}^{2^{j+1}-1}{n}^{-3\mathrm{/}2}\le 2^{-j\mathrm{/}2}$ bounds the tail by a geometric series.

By [F4] and step 3.1, almost surely $G_n$ fails for all sufficiently large $n$; by step 2.1 this means that almost surely the path has no finite derivative with absolute value at most $C$ at any point of $[a,b]$ (two-sided on $(a,b)$, right at $a$, left at $b$).

Intersect the common continuity event from [F1] with the complements of all the measurable limsup events of [F4]. Taking the union of those null events over the countably many rational pairs $0\le a<b$ and over integers $C\ge 1$, and using [F5] to place every $s>0$ in the interior of such an interval (while $s=0$ is the left endpoint of one), we obtain: almost surely no time $s\ge 0$ has a finite two-sided derivative (for $s>0$) or finite right derivative (for $s=0$).

The boundary cases are covered by the block choices of step 2.1: $s=0$ uses the right-handed block beginning at $a$, $s=b$ the left-handed block ending at $b$, and interior times either the forward or the backward block, all of which stay inside $[a,b]$; the cases n<6 are defined to be empty events in step 3.1, so the sequence is indexed by all natural numbers and no division by zero is performed; rounding the derivative bound up to an integer $C$ loses nothing, and the finite-difference ratio of [F2] is the definition-level form of The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set; AC is inherited through [F6] from the Brownian and normal-law interfaces.