Can a regular language contain non regular strings?

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.

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.
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.
• But the problem says that $u$ belongs to $\{0,1\}^*$, accordidng to my interpretation, for every u belonging to this set the resulting string should be regular. – Carlitos_30 Apr 27 at 15:52