Solution
= Solution
The scaled process starts at zero and has <continuous> paths. Disjoint time intervals map to disjoint intervals under $t\mapsto c^2t$, preserving independence of increments, and
$$
\frac{B_{c^2t}-B_{c^2s}}c\sim N\left(0,\frac{c^2(t-s)}{c^2}\right)=N(0,t-s).
$$
This holds for either sign of $c\ne0$. Hence $\boxed{\widetilde B_t=c^{-1}B_{c^2t}\text{ is standard Brownian motion}}$.