Suppose we have some non-regular context free language L. Suppose we also have language of all prefixes of words in L.

What can be an example of non-regular language L such that language of it’s prefixes is regular (Can be represented by a finite automaton)?

I don’t understand how language of prefixes can ever be regular, since the set of prefixes of a word include the word itself.

For example $ L= a^nb^n$ is my non-regular language. The language of it’s prefixes would include : $ \epsilon,a^n$ where $ n\ge 1$ ,$ a^nb$ where $ n\ge 1$ etc…

But what about b’s ? We need to know how many a’s there were in the first place. Therefore I don’t see how the language of prefixes can be regular.