Can a regular language contain non regular strings?

Question

Check this problem (1.71 from Sipser 3rd edition):

Let $\sum = \{0,1\}$. Let $A =\{0^ku0^k \ | \ k \ge 1 \ and \ u \in \sum^*$}. Show that $A$ is regular.

$u$ can be $\{0,00,000,...,01,011,...,1,11,111,11111...etc\}$

So, the language $0^k10^k \ with \ k\geq1$ should be regular, $u=1 \in \sum^*$, but is not regular according to pumping lemma, or because you need to count infinite states at the start and end.

I thought a regular language was a set of strings which every string being recognizable by a finite automata.

Answer 1

Well, the language $L=\{0^k10^m\mid k,m\in{\Bbb N}_0\}$ is regular, but the subset $\{0^k10^k\mid k\in{\Bbb N}_0\}$ is not. The accepting automaton for $L$ doesn't care how many $0$'s after the prefix $0^k1$ follow.

Answer 2

It's not necessary that the subset of a regular language should also be regular. In your example, observe that any string starting and ending with $0$ belongs to the language, and any other string does not.