= Solution
{c}
Each half-line $(-\infty,x]$ is a closed <Borel set>, and the empty set is Borel. Thus $\sigma(\pi(\mathbb R))\subseteq\mathcal B(\mathbb R)$.
Conversely,
$$
(-\infty,b)=\bigcup_{n\geq1}(-\infty,b-1/n],\qquad
(a,b)=(-\infty,b)\setminus(-\infty,a].
$$
The generated <sigma-algebra> contains both sets on the right and is closed under their difference, so it contains every open interval. By the given definition of the <Borel sigma-algebra>, the reverse inclusion follows. Therefore
$$
\boxed{\sigma(\pi(\mathbb R))=\mathcal B(\mathbb R).}
$$
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