Example of non-regular context free language L such that prefix(L) is regular 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.
Consider the language of words with the same number of a's and b's. It is non-regular but the set of prefixes is the set of all words which clearly is regular.