Suppose that we have two alphabets $\{$a$,$b$,$c$\}$ and $\{0,1\},$ and we have a function, where f$($a$)=$f$($b$)=0$ and f$($c$)=1.$ That is$,$ f maps a and b to $0$ and c to $1.$ Now suppose that we extend f to F: in the natural way: F$($w$)=$F$($w$1,$w$2,...$wn$)=$f$($w$1)$f$($w$2)...$f$($wn$).$ That is$,$ F maps a string symbol by symbols using f$.$ For example, F$($aabbca$)=$f$($a$)$f$($a$)$f$($b$)$f$($b$)$f$($c$)$f$($a$)=000010$ Is it true that if language L is regular, then so is F$($L$)?$

Yes, always

No$,$ it fails to hold for some regular languages

Only for finite languages

Only for infinite languages