# Is a particular string regular (e.g is '010') regular?

If the alphabet is $$\{0,1\}$$, then is the string '010' regular?

I think it is regular because DFA and regular languages are equivalent and this string has a DFA but at the same time it seems to contradict pumping lemma which implies not regular. Here is what I mean.

For pumping lemma first we have to take $$w$$ only in the language (here the language is just '010') So $$w=$$ '010'. I choose $$k = 2$$ then the new string $$w=xy^kz$$ has length more than 3 so it is definitely not '010', which means this language is not regular!

What am I missing?

However, it seems that by '010' you really mean the language consisting only of the single word '010', that is, $$\{ 010 \}$$. This language, just like any other finite language, is regular.
If $$L$$ is a regular language then there exists a constant $$n$$ such that for all words $$w \in L$$ of length at least $$n$$ there is a decomposition $$w = xyz$$, where $$|xy| \leq n$$ and $$y \neq \epsilon$$, such that $$xy^iz \in L$$ for all $$i \geq 0$$.
Your argument is ignoring the constant $$n$$, which could be larger than the length of all words in $$L$$.