Regularity of “middles” of words from regular language
Let $R$ be a regular language. Then there is a finite automaton $M$ for which the language, $L(M)$, of $M$ is $R$. Without loss of generality, we may assume that $M$ has no unreachable states. Construct a new FA, $M'$, as follows:


For each state $q$ of $M$ which has a sequence of transitions to a final state of $M$, make $q$ a final state. Call the set of such states $S$.
Add a new start state $q_0$ having $\epsilon$-transitions to each element of $S$.


It's not hard to see that $M'$ will accept any string which is a substring of some string in $R$.