= Solution
{c}
For positive times the transformed process is centered and jointly Gaussian, and
$$
\mathbb E(X_sX_t)=st\min(1/s,1/t)=\min(s,t).
$$
Its paths are continuous away from zero. To verify continuity at zero, we need $B_u/u\to0$ almost surely as $u\to\infty$. The <strong law of large numbers> applied to the independent increments $B_{n+1}-B_n$ gives $B_n/n\to0$. On the unit interval after $n$, the <Doob L2 maximal inequality> gives
$$
\mathbb P\left(\sup_{0\leq v\leq1}|B_{n+v}-B_n|>\varepsilon n\right)\leq\frac4{\varepsilon^2n^2}.
$$
The <Borel-Cantelli first lemma>, first for each positive rational $\varepsilon$, shows that these between-integer fluctuations divided by $n$ tend to zero. Hence $B_u/u\to0$ for all real $u\to\infty$, and $tB_{1/t}\to0$ as $t\downarrow0$.
The <Gaussian-process characterization of Brownian motion> now proves \b[$X$ is standard Brownian motion], with respect to its natural filtration. This is <time inversion of Brownian motion>; the transformed process need not be adapted to the original forward filtration.
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