Fixed-time assertions do not yield a pathwise nowhere statement

The profile is continuous and differentiable at the origin: $G(s)=s^{2}T(s)$ is a product of continuous functions, and with $M:={\sup }_{[0,1]}\mathrm{\mid }T\mathrm{\mid }<\mathrm{\infty }$, finite because the continuous $T$ is bounded on the compact interval, one has $\mathrm{\mid }G(s)\mathrm{\mid }\le M{s}^2$ for every $s\in [0,1]$, so $G(s)\mathrm{/}s\to 0$ and $G^{\mathrm{\prime }}(0)=0$.

The profile is nowhere else differentiable: if $G$ were differentiable at some $s\in (0,1]$, then $T=G\mathrm{/}{s}^2$ would be differentiable at $s$ as a quotient with nonvanishing denominator, contradicting the nowhere differentiability of the Takagi function on $[0,1]$.

Every sample path $t\mapsto X_t(\omega )=G(\mathrm{\mid }t-\omega \mathrm{\mid })$ is continuous, being a composition of the continuous maps $t\mapsto \mathrm{\mid }t-\omega \mathrm{\mid }$ and $G$; for the same reason the map $(t,\omega )\mapsto X_t(\omega )$ is jointly measurable.

At $t=\omega$ the path is differentiable with derivative $0$: for $h\mathrm{\ne }0$ the difference quotient is $(G(\mathrm{\mid }h\mathrm{\mid })-G(0))\mathrm{/}h=\pm G(\mathrm{\mid }h\mathrm{\mid })\mathrm{/}\mathrm{\mid }h\mathrm{\mid }$, whose limit as $h\to 0$ is $\pm G^{\mathrm{\prime }}(0)=0$ by [step 1.1].

At every $t\in (0,1)$ with $t\mathrm{\ne }\omega$ the path is not differentiable at $t$: on a neighbourhood of such a $t$ that lies inside $(0,1)$ the path is the composition of the affine map $u\mapsto \pm (u-\omega )$ of nonzero slope with the restriction of $G$, so differentiability of the path at $t$ would make $G=X\circ A^{-1}$ differentiable at $\mathrm{\mid }t-\omega \mathrm{\mid }$, which lies in $(0,1)$ because $t\in (0,1)$ and $\omega \in [0,1]$, and [step 2.1] excludes exactly that.

For a fixed deterministic $t\in (0,1)$ the path is differentiable at $t$ exactly when $\omega =t$, by [step 2.3] and [step 3.1]; since the singleton $\{t\}$ is Lebesgue-null, the path is almost surely not differentiable at $t$, and this holds for every $t$ of the uncountable family $(0,1)$.

Yet almost surely the path is differentiable somewhere, namely at $t=\omega \in (0,1)$, by [step 2.3]; hence the pathwise event "the path is nowhere differentiable on $(0,1)$" has probability $0$. The fixed-time assertions of [step 4.1] therefore do not upgrade to the pathwise statement: the quantifier over the uncountable family of times cannot be moved inside the almost-sure statement, which is the defect being exhibited.

The boundary and degenerate cases are covered: the fixed-time family is the open interval $(0,1)$, so that at each of its times the two-sided notion of differentiability applies and the endpoint behaviour of the profile is never needed; the case $t=\omega$ is the differentiability point of the path and contributes the null singleton to the fixed-time computation of [step 4.1]; the outcomes $\omega \in \{0,1\}$ form a null edge case and are not needed, since [step 5.1] only requires the event of positive probability on which the path is differentiable at an interior time; the profile is not constant, so the degenerate case in which every time were a differentiability point does not arise; and the construction selects nothing, the measure space and the profile being explicit.